This document discusses different types of retaining walls, including:
- Gravity walls, pre-cast crib walls, gabion walls, reinforced concrete walls, sheet pile walls, mechanically stabilized earth (MSE) walls, slurry walls, secant pile walls, soldier piles and lagging walls, cofferdam walls, and hybrid systems.
It provides details on the materials, designs, and uses of various retaining wall types. Common materials include wood, steel, concrete, and soil reinforcements. Walls are chosen based on factors like height, site conditions, costs, and whether they are temporary or permanent.
The document discusses response spectra, which are plots of the maximum response of single-degree-of-freedom oscillators versus their natural period when subjected to a specific ground motion. Response spectra allow characterization of earthquake ground motions and are commonly used in earthquake-resistant design. The analysis procedure using response spectra involves obtaining the spectral acceleration from the response spectrum curve based on an oscillator's period and calculating the maximum displacement, base shear, and overturning moment. This procedure can be extended to multi-degree-of-freedom structures. An example is provided to demonstrate using a response spectrum to determine interstory drifts and story shears in a two-story building.
This document reports on a study investigating the shielding properties of various wood species and concrete materials commonly used in Ghana. Samples of five wood species - Triplochiton Scleroxylon, Khaya Ivorensis, Milicia excelsa, Guibourtia Ehie, Albizia Ferruginea and Terminilia Superba - and three types of concrete made with different cements were tested. The mass attenuation coefficients of the materials were measured using a broad beam x-ray setup at energies of 80-120kV. The results showed that Triplochiton Scleroxylon had the highest mass attenuation coefficients for wood, while Ghacem cement concrete had the highest coefficients for concrete. This
Compaction characteristics of fine grained soilavirup naskar
The document discusses compaction of soils. It defines compaction as artificially rearranging and packing soil particles into a closer state through mechanical means to decrease porosity and increase dry density. Compaction is done for purposes like increasing density, strength, load bearing capacity, and stability while decreasing compressibility, permeability, and erosion damage. It reviews literature on field permeability tests being more accurate than lab tests, correlating compaction characteristics like optimum moisture content with thermal behavior, and stabilizing compacted clay through admixtures or compactive effort. The conclusion discusses the importance of field tests, avoiding thin clay liners, compacting wet of optimum, relationships between density, moisture content and thermal properties, not rejecting high saturation tests,
The document provides steps to calculate the centerline length of a brick wall for quantity take-off purposes. It explains that the centerline length is needed rather than the external perimeter to accurately measure quantities. It describes subtracting half the wall width from each corner to calculate the centerline from the external perimeter. Alternatively, it shows adding half the wall width to each corner when calculating the centerline from the internal perimeter. The area of the brickwork can then be found by multiplying the centerline length by the wall height.
Solution Strategies for Equations that Arise in Geometric (Clifford) AlgebraJames Smith
Drawing mainly upon exercises from Hestenes's New Foundations for Classical Mechanics, this document presents, explains, and discusses common solution strategies. Included are a list of formulas and a guide to nomenclature.
See also:
http://www.slideshare.net/JamesSmith245/rotations-of-vectors-via-geometric-algebra-explanation-and-usage-in-solving-classic-geometric-construction-problems-version-of-11-february-2016 ;
http://www.slideshare.net/JamesSmith245/resoluciones-de-problemas-de-construccin-geomtricos-por-medio-de-la-geometra-clsica-y-el-lgebra-geomtrica-vectorial ;
http://www.slideshare.net/JamesSmith245/solution-of-the-special-case-clp-of-the-problem-of-apollonius-via-vector-rotations-using-geometric-algebra ;
http://www.slideshare.net/JamesSmith245/solution-of-the-ccp-case-of-the-problem-of-apollonius-via-geometric-clifford-algebra ;
http://www.slideshare.net/JamesSmith245/a-very-brief-introduction-to-reflections-in-2d-geometric-algebra-and-their-use-in-solving-construction-problems ;
http://www.slideshare.net/JamesSmith245/solution-of-the-llp-limiting-case-of-the-problem-of-apollonius-via-geometric-algebra-using-reflections-and-rotations ;
http://www.slideshare.net/JamesSmith245/simplied-solutions-of-the-clp-and-ccp-limiting-cases-of-the-problem-of-apollonius-via-vector-rotations-using-geometric-algebra ;
http://www.slideshare.net/JamesSmith245/additional-solutions-of-the-limiting-case-clp-of-the-problem-of-apollonius-via-vector-rotations-using-geometric-algebra ;
http://www.slideshare.net/JamesSmith245/an-additional-brief-solution-of-the-cpp-limiting-case-of-the-problem-of-apollonius-via-geometric-algebra-ga .
Construcciones para encontrar la raíz cuadrada y resolver ecuaciones cuadráticasJames Smith
Se presentan y explican construcciones geométricas (realizadas con regla y compás) para encontrar la raíz cuadrada de un número, y resolver ecuaciones cuadráticas.
This document discusses different types of retaining walls, including:
- Gravity walls, pre-cast crib walls, gabion walls, reinforced concrete walls, sheet pile walls, mechanically stabilized earth (MSE) walls, slurry walls, secant pile walls, soldier piles and lagging walls, cofferdam walls, and hybrid systems.
It provides details on the materials, designs, and uses of various retaining wall types. Common materials include wood, steel, concrete, and soil reinforcements. Walls are chosen based on factors like height, site conditions, costs, and whether they are temporary or permanent.
The document discusses response spectra, which are plots of the maximum response of single-degree-of-freedom oscillators versus their natural period when subjected to a specific ground motion. Response spectra allow characterization of earthquake ground motions and are commonly used in earthquake-resistant design. The analysis procedure using response spectra involves obtaining the spectral acceleration from the response spectrum curve based on an oscillator's period and calculating the maximum displacement, base shear, and overturning moment. This procedure can be extended to multi-degree-of-freedom structures. An example is provided to demonstrate using a response spectrum to determine interstory drifts and story shears in a two-story building.
This document reports on a study investigating the shielding properties of various wood species and concrete materials commonly used in Ghana. Samples of five wood species - Triplochiton Scleroxylon, Khaya Ivorensis, Milicia excelsa, Guibourtia Ehie, Albizia Ferruginea and Terminilia Superba - and three types of concrete made with different cements were tested. The mass attenuation coefficients of the materials were measured using a broad beam x-ray setup at energies of 80-120kV. The results showed that Triplochiton Scleroxylon had the highest mass attenuation coefficients for wood, while Ghacem cement concrete had the highest coefficients for concrete. This
Compaction characteristics of fine grained soilavirup naskar
The document discusses compaction of soils. It defines compaction as artificially rearranging and packing soil particles into a closer state through mechanical means to decrease porosity and increase dry density. Compaction is done for purposes like increasing density, strength, load bearing capacity, and stability while decreasing compressibility, permeability, and erosion damage. It reviews literature on field permeability tests being more accurate than lab tests, correlating compaction characteristics like optimum moisture content with thermal behavior, and stabilizing compacted clay through admixtures or compactive effort. The conclusion discusses the importance of field tests, avoiding thin clay liners, compacting wet of optimum, relationships between density, moisture content and thermal properties, not rejecting high saturation tests,
The document provides steps to calculate the centerline length of a brick wall for quantity take-off purposes. It explains that the centerline length is needed rather than the external perimeter to accurately measure quantities. It describes subtracting half the wall width from each corner to calculate the centerline from the external perimeter. Alternatively, it shows adding half the wall width to each corner when calculating the centerline from the internal perimeter. The area of the brickwork can then be found by multiplying the centerline length by the wall height.
Solution Strategies for Equations that Arise in Geometric (Clifford) AlgebraJames Smith
Drawing mainly upon exercises from Hestenes's New Foundations for Classical Mechanics, this document presents, explains, and discusses common solution strategies. Included are a list of formulas and a guide to nomenclature.
See also:
http://www.slideshare.net/JamesSmith245/rotations-of-vectors-via-geometric-algebra-explanation-and-usage-in-solving-classic-geometric-construction-problems-version-of-11-february-2016 ;
http://www.slideshare.net/JamesSmith245/resoluciones-de-problemas-de-construccin-geomtricos-por-medio-de-la-geometra-clsica-y-el-lgebra-geomtrica-vectorial ;
http://www.slideshare.net/JamesSmith245/solution-of-the-special-case-clp-of-the-problem-of-apollonius-via-vector-rotations-using-geometric-algebra ;
http://www.slideshare.net/JamesSmith245/solution-of-the-ccp-case-of-the-problem-of-apollonius-via-geometric-clifford-algebra ;
http://www.slideshare.net/JamesSmith245/a-very-brief-introduction-to-reflections-in-2d-geometric-algebra-and-their-use-in-solving-construction-problems ;
http://www.slideshare.net/JamesSmith245/solution-of-the-llp-limiting-case-of-the-problem-of-apollonius-via-geometric-algebra-using-reflections-and-rotations ;
http://www.slideshare.net/JamesSmith245/simplied-solutions-of-the-clp-and-ccp-limiting-cases-of-the-problem-of-apollonius-via-vector-rotations-using-geometric-algebra ;
http://www.slideshare.net/JamesSmith245/additional-solutions-of-the-limiting-case-clp-of-the-problem-of-apollonius-via-vector-rotations-using-geometric-algebra ;
http://www.slideshare.net/JamesSmith245/an-additional-brief-solution-of-the-cpp-limiting-case-of-the-problem-of-apollonius-via-geometric-algebra-ga .
Construcciones para encontrar la raíz cuadrada y resolver ecuaciones cuadráticasJames Smith
Se presentan y explican construcciones geométricas (realizadas con regla y compás) para encontrar la raíz cuadrada de un número, y resolver ecuaciones cuadráticas.
A Very Brief Introduction to Reflections in 2D Geometric Algebra, and their U...James Smith
This document discusses reflections of vectors and of geometrical products of two vectors, in two-dimensional Geometric Algebra (GA). It then uses reflections to solve a simple tangency problem.
Additional Solutions of the Limiting Case "CLP" of the Problem of Apollonius ...James Smith
This document uses geometric algebra to solve the limiting case of the Problem of Apollonius known as the Circle-Line-Point problem. It presents three solutions: one using only rotations, one using a combination of reflections and rotations, and one in the appendix using only rotations. The solutions identify either the points of tangency between the solution circles and the given circle, or the points of tangency between the solution circles and the given line. The document reviews reflections and rotations in geometric algebra to establish the necessary foundations before presenting the solutions.
Calculation of the Curvature Expected in Photographs of a Sphere's HorizonJames Smith
A formula is derived for the curvature of the horizon's image in photos of a sphere of radius R , taken by a camera with horizontal view angle alpha from height h above the sphere's surface. The formula is validated by means of an interactive GeoGebra construction: a key angle calculated from the formula derived here is compared to the angle actually present in the construction. Using the validated formula, the amount of curvature expected to be present in a photo of the Earth's horizon from an altitude of 3 m is calculated. The result is an order of magnitude smaller than typical degrees of barrel distortion present in consumers' digital cameras. Therefore, claims that "flat horizons in photos of waterscapes prove that the Earth is flat" are untenable.
An additional brief solution of the CPP limiting case of the Problem of Apoll...James Smith
This document adds to the collection of GA solutions to plane-geometry problems, most of them dealing with tangency, that are presented in References 1-7. Reference 1 presented several ways of solving the CPP limiting case of the Problem of Apollonius. Here, we use ideas from Reference 6 to solve that case in yet another way.
5 Tips On How To Grow Your Salon BusinessSalon Genie
This document promotes trying something for free without any obligation. It does not provide any details on what specifically can be tried for free, but encourages the reader to take advantage of a free trial offer. The brief message focuses on allowing the reader to experience something at no cost upfront to see if they like it.
Why Does the Atmosphere Rotate? Trajectory of a desorbed moleculeJames Smith
As a step toward understanding why the Earth's atmosphere "rotates" with the Earth, we use using Geometric (Clifford) Algebra to investigate the trajectory of a single molecule that desorbs vertically upward from the Equator, then falls back to Earth without colliding with any other molecules. Sample calculations are presented for a molecule whose vertical velocity is equal to the surface velocity of the Earth at the Equator (463 m/s) and for one with a vertical velocity three times as high. The latter velocity is sufficient for the molecule to reach the Kármán Line (100,000 m). We find that both molecules fall to Earth behind the point from which they desorbed: by 0.25 degrees of latitude for the higher vertical velocity, but by only 0.001 degrees for the lower.
International Programmes in Thailand & ASEAN vol.9 part 2ธีระพล ชัยมงคลกานต์
Mahanakorn University of Technology provides information on its Bilingual Programme, which offers undergraduate programs in chemical, civil, electrical, and mechanical engineering taught in both Thai and English. The program aims to prepare students for the ASEAN Economic Community and global workplace. Students can study for two years at MUT and two years at partner universities in the UK, or study for four years entirely at MUT to earn a bachelor's degree. Admission requirements and program costs are provided. Contact information is given for those seeking more details.
International Programmes in Thailand & ASEAN vol.9 part 1ธีระพล ชัยมงคลกานต์
This document appears to be a volume of the publication "International Programs in THAILAND & ASEAN" that provides information about higher education opportunities in Thailand and Southeast Asia. It includes sections about Thai and ASEAN higher education policies, profiles of universities in Thailand that offer international programs, and information about the ASEAN University Network. The volume aims to be an informative resource for students considering higher education options in the region. It is distributed widely in Thailand, Southeast Asia, and internationally to help students make informed decisions about university choices.
Sense/Net ECM is the industry’s first open source Microsoft SharePoint alternative for the .Net platform. Sense/Net ECM is dual licensed, with a free Community edition, and a fully supported Enterprise edition. It is sold via solution partners who design and build collaborative intranets, extranets, and public sites for their clients.
The coordinate plane is formed by intersecting two number lines, called the x-axis and y-axis, at their zero points. The point of intersection is called the origin. To graph an inequality in two variables, graph the boundary curve and shade the region where the inequality is true. The distance formula can be used to find the distance between two points by treating it as the hypotenuse of a right triangle formed by the differences in the x and y coordinates. The midpoint formula finds the point halfway between two points by averaging the x and y coordinates. A circle is defined as all points equidistant from a center point, where the distance from the center is the radius. The standard form of a circle equation relates the
1. The assignment can be completed in groups of 4 students and must be typed using Microsoft Word and Equation Editor. References from at least 2 books must be used in addition to internet sources.
2. L'Hospital's Rule can be used to evaluate limits involving indeterminate forms such as 0/0, ∞/∞, and other cases. It uses derivatives to help find the limit.
3. Conic sections are curves formed by the intersection of a plane with a double cone. The five types are a pair of intersecting lines, a circle, a parabola, an ellipse, and a hyperbola. Each has a specific formation method and properties that can be used to graph
A cone is a 3D geometric shape that tapers from a flat circular base to a single point called the apex. The formulas for the volume and surface area of a cone involve the radius of the base and the height or slant height. Word problems apply these formulas to calculate missing values like height, volume, or surface area when other values like radius or volume are given.
This document provides an overview of chapter six which discusses applications of the definite integral in geometry, science, and engineering. It introduces how definite integrals can be used to calculate volume, surface area, length of a plane curve, and work done by a force. It reviews key concepts like Riemann sums and finding the area between two curves. It then explains the specific applications of using integrals to find volume of solids obtained by rotating an area about an axis, surface area of revolution, and work done by a variable force. Examples are provided for each application.
A circular cone is a surface generated by a straight line passing through a fixed point and moving on a circle. A right circular cone is generated by revolving a straight line that passes through a fixed point and makes a constant angle with a fixed line. The height of a cone is the vertical distance between the vertex and base. The slant height is the distance between the vertex and any point on the base circumference. The volume of a right circular cone is 1/3πr^2h, where r is the radius and h is the height. An example calculates the volume of a cone formed by rolling up a sector of a circle.
This document provides information about calculating the surface area and volume of prisms and cylinders. It defines right and oblique prisms, and explains that the volume formula is the same for both as V=Bh, where B is the base area and h is the altitude or height. The same is true for cylinders, where the volume is V=πr^2h. Examples are given to find the volume of various prisms and cylinders. Formulas are also provided for calculating the lateral and total surface area of prisms as S=L+2B, where L is the lateral area calculated as L=Ph, and B is the base area. For cylinders, the lateral area formula is L=C*h,
1. The document defines key terms related to circles such as diameter, radius, chord, arc, and sector.
2. Several theorems about circles are presented, including that equal chords of a circle subtend equal angles at the center, and the perpendicular from the center of a circle to a chord bisects the chord.
3. The document summarizes that a circle can be defined as all points equidistant from a fixed point, and introduces various properties and relationships regarding angles, chords, and points on circles.
This report summarizes research on the motion of particles on curves. It was found that:
1) The center of mass of 3 points on an ellipse that divide its perimeter evenly traces out a smaller ellipse of the same shape.
2) The maximum product of distances between 4 particles on a rectangle occurs when particles are at the corners for small rectangles, but 2 particles move off the corners for larger rectangles.
3) The center of mass of n points on a square that divide its perimeter evenly traces out a smaller square n times for odd n, and remains fixed at the center for even n.
En esté trabajo vamos a ver que las transformaciones se pueden considerar como sistemas de coordenadas definidos respecto a un sistema de coordenadas global, y veremos la ventaja de comprender esta dualidad en el ámbito de los gráficos por ordenador
A Very Brief Introduction to Reflections in 2D Geometric Algebra, and their U...James Smith
This document discusses reflections of vectors and of geometrical products of two vectors, in two-dimensional Geometric Algebra (GA). It then uses reflections to solve a simple tangency problem.
Additional Solutions of the Limiting Case "CLP" of the Problem of Apollonius ...James Smith
This document uses geometric algebra to solve the limiting case of the Problem of Apollonius known as the Circle-Line-Point problem. It presents three solutions: one using only rotations, one using a combination of reflections and rotations, and one in the appendix using only rotations. The solutions identify either the points of tangency between the solution circles and the given circle, or the points of tangency between the solution circles and the given line. The document reviews reflections and rotations in geometric algebra to establish the necessary foundations before presenting the solutions.
Calculation of the Curvature Expected in Photographs of a Sphere's HorizonJames Smith
A formula is derived for the curvature of the horizon's image in photos of a sphere of radius R , taken by a camera with horizontal view angle alpha from height h above the sphere's surface. The formula is validated by means of an interactive GeoGebra construction: a key angle calculated from the formula derived here is compared to the angle actually present in the construction. Using the validated formula, the amount of curvature expected to be present in a photo of the Earth's horizon from an altitude of 3 m is calculated. The result is an order of magnitude smaller than typical degrees of barrel distortion present in consumers' digital cameras. Therefore, claims that "flat horizons in photos of waterscapes prove that the Earth is flat" are untenable.
An additional brief solution of the CPP limiting case of the Problem of Apoll...James Smith
This document adds to the collection of GA solutions to plane-geometry problems, most of them dealing with tangency, that are presented in References 1-7. Reference 1 presented several ways of solving the CPP limiting case of the Problem of Apollonius. Here, we use ideas from Reference 6 to solve that case in yet another way.
5 Tips On How To Grow Your Salon BusinessSalon Genie
This document promotes trying something for free without any obligation. It does not provide any details on what specifically can be tried for free, but encourages the reader to take advantage of a free trial offer. The brief message focuses on allowing the reader to experience something at no cost upfront to see if they like it.
Why Does the Atmosphere Rotate? Trajectory of a desorbed moleculeJames Smith
As a step toward understanding why the Earth's atmosphere "rotates" with the Earth, we use using Geometric (Clifford) Algebra to investigate the trajectory of a single molecule that desorbs vertically upward from the Equator, then falls back to Earth without colliding with any other molecules. Sample calculations are presented for a molecule whose vertical velocity is equal to the surface velocity of the Earth at the Equator (463 m/s) and for one with a vertical velocity three times as high. The latter velocity is sufficient for the molecule to reach the Kármán Line (100,000 m). We find that both molecules fall to Earth behind the point from which they desorbed: by 0.25 degrees of latitude for the higher vertical velocity, but by only 0.001 degrees for the lower.
International Programmes in Thailand & ASEAN vol.9 part 2ธีระพล ชัยมงคลกานต์
Mahanakorn University of Technology provides information on its Bilingual Programme, which offers undergraduate programs in chemical, civil, electrical, and mechanical engineering taught in both Thai and English. The program aims to prepare students for the ASEAN Economic Community and global workplace. Students can study for two years at MUT and two years at partner universities in the UK, or study for four years entirely at MUT to earn a bachelor's degree. Admission requirements and program costs are provided. Contact information is given for those seeking more details.
International Programmes in Thailand & ASEAN vol.9 part 1ธีระพล ชัยมงคลกานต์
This document appears to be a volume of the publication "International Programs in THAILAND & ASEAN" that provides information about higher education opportunities in Thailand and Southeast Asia. It includes sections about Thai and ASEAN higher education policies, profiles of universities in Thailand that offer international programs, and information about the ASEAN University Network. The volume aims to be an informative resource for students considering higher education options in the region. It is distributed widely in Thailand, Southeast Asia, and internationally to help students make informed decisions about university choices.
Sense/Net ECM is the industry’s first open source Microsoft SharePoint alternative for the .Net platform. Sense/Net ECM is dual licensed, with a free Community edition, and a fully supported Enterprise edition. It is sold via solution partners who design and build collaborative intranets, extranets, and public sites for their clients.
The coordinate plane is formed by intersecting two number lines, called the x-axis and y-axis, at their zero points. The point of intersection is called the origin. To graph an inequality in two variables, graph the boundary curve and shade the region where the inequality is true. The distance formula can be used to find the distance between two points by treating it as the hypotenuse of a right triangle formed by the differences in the x and y coordinates. The midpoint formula finds the point halfway between two points by averaging the x and y coordinates. A circle is defined as all points equidistant from a center point, where the distance from the center is the radius. The standard form of a circle equation relates the
1. The assignment can be completed in groups of 4 students and must be typed using Microsoft Word and Equation Editor. References from at least 2 books must be used in addition to internet sources.
2. L'Hospital's Rule can be used to evaluate limits involving indeterminate forms such as 0/0, ∞/∞, and other cases. It uses derivatives to help find the limit.
3. Conic sections are curves formed by the intersection of a plane with a double cone. The five types are a pair of intersecting lines, a circle, a parabola, an ellipse, and a hyperbola. Each has a specific formation method and properties that can be used to graph
A cone is a 3D geometric shape that tapers from a flat circular base to a single point called the apex. The formulas for the volume and surface area of a cone involve the radius of the base and the height or slant height. Word problems apply these formulas to calculate missing values like height, volume, or surface area when other values like radius or volume are given.
This document provides an overview of chapter six which discusses applications of the definite integral in geometry, science, and engineering. It introduces how definite integrals can be used to calculate volume, surface area, length of a plane curve, and work done by a force. It reviews key concepts like Riemann sums and finding the area between two curves. It then explains the specific applications of using integrals to find volume of solids obtained by rotating an area about an axis, surface area of revolution, and work done by a variable force. Examples are provided for each application.
A circular cone is a surface generated by a straight line passing through a fixed point and moving on a circle. A right circular cone is generated by revolving a straight line that passes through a fixed point and makes a constant angle with a fixed line. The height of a cone is the vertical distance between the vertex and base. The slant height is the distance between the vertex and any point on the base circumference. The volume of a right circular cone is 1/3πr^2h, where r is the radius and h is the height. An example calculates the volume of a cone formed by rolling up a sector of a circle.
This document provides information about calculating the surface area and volume of prisms and cylinders. It defines right and oblique prisms, and explains that the volume formula is the same for both as V=Bh, where B is the base area and h is the altitude or height. The same is true for cylinders, where the volume is V=πr^2h. Examples are given to find the volume of various prisms and cylinders. Formulas are also provided for calculating the lateral and total surface area of prisms as S=L+2B, where L is the lateral area calculated as L=Ph, and B is the base area. For cylinders, the lateral area formula is L=C*h,
1. The document defines key terms related to circles such as diameter, radius, chord, arc, and sector.
2. Several theorems about circles are presented, including that equal chords of a circle subtend equal angles at the center, and the perpendicular from the center of a circle to a chord bisects the chord.
3. The document summarizes that a circle can be defined as all points equidistant from a fixed point, and introduces various properties and relationships regarding angles, chords, and points on circles.
This report summarizes research on the motion of particles on curves. It was found that:
1) The center of mass of 3 points on an ellipse that divide its perimeter evenly traces out a smaller ellipse of the same shape.
2) The maximum product of distances between 4 particles on a rectangle occurs when particles are at the corners for small rectangles, but 2 particles move off the corners for larger rectangles.
3) The center of mass of n points on a square that divide its perimeter evenly traces out a smaller square n times for odd n, and remains fixed at the center for even n.
En esté trabajo vamos a ver que las transformaciones se pueden considerar como sistemas de coordenadas definidos respecto a un sistema de coordenadas global, y veremos la ventaja de comprender esta dualidad en el ámbito de los gráficos por ordenador
The document describes key concepts related to the Cartesian plane including:
- The Cartesian plane consists of two perpendicular axes (x and y) intersecting at the origin point.
- Points on the plane are represented as ordered pairs (x,y).
- The distance between two points P1(x1,y1) and P2(x2,y2) is given by the formula d = √(x2 - x1)2 + (y2 - y1)2.
- Circles, parabolas, ellipses, and hyperbolas are examples of curves that can be represented on the Cartesian plane using algebraic equations. Their properties and equations are discussed.
1. The document defines norms and normed spaces, which allow the measurement of vector length. This enables the calculation of shapes' circumferences.
2. Formulas are provided to calculate the clockwise and counterclockwise circumferences of a rectangular metric space. Both formulas sum to the same length.
3. Formulas are also given for calculating the circumferences of non-symmetric quadrilaterals. An example shows the circumferences can differ depending on direction, proving an asymmetric metric space exists.
Circle is a simple closed shape in Euclidean geometry. It is defined as the set of all points in a plane that are equidistant from a given point, called the center. The distance from the center to any point on the circle is called the radius. A circle can also be defined as a special type of ellipse where the two foci are coincident or as the shape that encloses the maximum area for a given perimeter. Key properties of circles include relationships between circumference, diameter, radius and area. Tangents, chords, and inscribed angles also have important properties related to circles.
The document provides a summary of coordinate geometry. It begins with definitions of key terms like the coordinate plane, axes, quadrants, and coordinates. It then discusses finding the midpoint, distance, and section formula between two points. Methods for finding the coordinates of the centroid and area of a triangle are presented. The document outlines different forms of equations for straight lines, including their slopes and the general equation of a line. It concludes with some uses of coordinate geometry, such as determining if lines are parallel/perpendicular.
The document discusses different types of cones, including right circular cones. It defines key terms like height, slant height, radius, and derives the formula for the volume of a right circular cone as 1/3 * π * r^2 * h. It provides examples of using the formula to calculate volumes of cones formed in different ways.
This document summarizes key concepts about circles. It defines circles and related terms like radius, diameter, chord, arc, and sector. It presents 8 theorems about angles subtended by chords and arcs, perpendiculars from the center to chords, circles through 3 points, equal chords and their distances from the center, and cyclic quadrilaterals. The concluding section summarizes that equal chords and arcs have corresponding relationships, angles in the same segment are equal, and properties of cyclic quadrilaterals. The document provides definitions, proofs, and conclusions about geometric properties of circles.
The document discusses circles, defining them as sets of points equidistant from a center point. It describes key circle terms like diameter, radius, chord, and circumference. Formulas are provided relating circumference to diameter using pi, diameter to radius, and area to radius. Examples demonstrate calculating circumference from diameter, diameter from circumference, and area from radius using the formulas. The document aims to define and explain key geometric concepts relating to circles through definitions, explanations, and example calculations.
The document discusses circles, defining them as sets of points equidistant from a center point. It describes key circle terms like diameter, radius, chord, and circumference. Formulas are provided relating circumference to diameter using pi, diameter to radius, and area to radius. Examples demonstrate calculating circumference from diameter, diameter from circumference, and area from radius using the formulas. The document aims to define and explain key geometric concepts relating to circles through definitions, explanations, and example calculations.
Lines and circles are one of the primary things you figure out how to draw when you start with
arithmetic in rudimentary classes. Notwithstanding, these basic figures bring more to the table than
what meets the eye.
This document provides an overview of algebra, trigonometry, and analytic geometry. It defines key concepts like functions, coordinate systems, Venn diagrams, and trigonometric functions. Functions are introduced as relations where each element of the domain corresponds to one and only one element in the range. Coordinate systems like the Cartesian plane are explained. Trigonometric functions like sine, cosine, and tangent are defined based on right triangles and the unit circle. Their domains and ranges are described along with periodic properties. Examples of trigonometric and other function types are also given.
The document provides information about circles including definitions, properties, theorems and history. It defines a circle as a simple closed curve where all points are equidistant from the center. Key properties discussed are that a circle's circumference and radius are proportional, and its area is proportional to the square of the radius. Theorems covered relate to chords, tangents, secants and inscribed angles. The document also discusses squaring the circle problem and circles in history from ancient Greeks to modern mathematics.
Similar to How to Calculate Distances from Centerline to Inside Walls of Domes, for Any Given Height above the Ground (20)
Using a Common Theme to Find Intersections of Spheres with Lines and Planes v...James Smith
After reviewing the sorts of calculations for which Geometric Algebra (GA) is especially convenient, we identify a common theme through which those types of calculations can be used to find the intersections of spheres with lines, planes, and other spheres.
Via Geometric Algebra: Direction and Distance between Two Points on a Spheric...James Smith
As a high-school-level example of solving a problem via Geometric (Clifford) Algebra, we show how to calculate the distance and direction between two points on Earth, given the locations' latitudes and longitudes. We validate the results by comparing them to those obtained from online calculators. This example invites a discussion of the benefits of teaching spherical trigonometry (the usual way of solving such problems) at the high-school level versus teaching how to use Geometric Algebra for the same purpose.
Solution of a Vector-Triangle Problem Via Geometric (Clifford) AlgebraJames Smith
As a high-school-level application of Geometric Algebra (GA), we show how to solve a simple vector-triangle problem. Our method highlights the use of outer products and inverses of bivectors.
Via Geometric (Clifford) Algebra: Equation for Line of Intersection of Two Pl...James Smith
As a high-school-level example of solving a problem via Geometric Algebra (GA), we show how to derive an equation for the line of intersection between two given planes. The solution method that we use emphasizes GA's capabilities for expressing and manipulating projections and rotations of vectors.
Solution of a Sangaku ``Tangency" Problem via Geometric AlgebraJames Smith
Because the shortage of worked-out examples at introductory levels is an obstacle to widespread adoption of Geometric Algebra (GA), we use GA to solve one of the beautiful \emph{sangaku} problems from 19th-Century Japan. Among the GA operations that prove useful is the rotation of vectors via the unit bivector
Un acercamiento a los determinantes e inversos de matricesJames Smith
Este documento presenta un resumen de tres oraciones sobre los determinantes e inversos de matrices. Introduce los conceptos de matrices y sistemas de ecuaciones lineales, y explica cómo la resolución de sistemas lleva a las ideas de determinantes de matrices y la inversa de una matriz. Finalmente, compara las versiones matricial y no matricial de resolver sistemas lineales.
Understanding the "Chain Rule" for Derivatives by Deriving Your Own VersionJames Smith
Because the Chain Rule can confuse students as much as it helps them solve real problems, we put ourselves in the shoes of the mathematicians who derived it, so that students may understand the motivation for the rule; its limitations; and why textbooks present it in its customary form. We begin by finding the derivative of sin2x without using the Chain Rule. That exercise, having shown that even a comparatively simple compound function can be bothersome to differentiate using the definition of the derivative as a limit, provides the motivation for developing our own formula for the derivative of the general compound function g[f(x)]. In the course of that development, we see why the function f must be continuous at any value of x to which the formula is applied. We finish by comparing our formula to that which is commonly given.
As a demonstration of the coherence of Geometric Algebra's (GA's) geometric and algebraic concepts of bivectors, we add three geometric bivectors according to the procedure described by Hestenes and Macdonald, then use bivector identities to determine, from the result, two vectors whose outer product is equal to the initial sum. In this way, we show that the procedure that GA's inventors defined for adding geometric bivectors is precisely that which is needed to give results that coincide with those obtained by calculating outer products of vectors that are expressed in terms of a 3D basis. We explain that that accomplishment is no coincidence: it is a consequence of the attributes that GA's designers assigned (or didn't) to bivectors.
Learning Geometric Algebra by Modeling Motions of the Earth and Shadows of Gn...James Smith
Because the shortage of worked-out examples at introductory levels is an obstacle to widespread adoption of Geometric Algebra (GA), we use GA to calculate Solar azimuths and altitudes as a function of time via the heliocentric model. We begin by representing the Earth's motions in GA terms. Our representation incorporates an estimate of the time at which the Earth would have reached perihelion in 2017 if not affected by the Moon's gravity. Using the geometry of the December 2016 solstice as a starting point, we then employ GA's capacities for handling rotations to determine the orientation of a gnomon at any given latitude and longitude during the period between the December solstices of 2016 and 2017. Subsequently, we derive equations for two angles: that between the Sun's rays and the gnomon's shaft, and that between the gnomon's shadow and the direction ``north" as traced on the ground at the gnomon's location. To validate our equations, we convert those angles to Solar azimuths and altitudes for comparison with simulations made by the program Stellarium. As further validation, we analyze our equations algebraically to predict (for example) the precise timings and locations of sunrises, sunsets, and Solar zeniths on the solstices and equinoxes. We emphasize that the accuracy of the results is only to be expected, given the high accuracy of the heliocentric model itself, and that the relevance of this work is the efficiency with which that model can be implemented via GA for teaching at the introductory level. On that point, comments and debate are encouraged and welcome.
Solution of a High-School Algebra Problem to Illustrate the Use of Elementary...James Smith
This document is the first in what is intended to be a collection of solutions of high-school-level problems via Geometric Algebra (GA). GA is very much "overpowered" for such problems, but students at that level who plan to go into more-advanced math and science courses will benefit from seeing how to "see" basic problems in GA terms, and to then solve those problems using GA identities and common techniques.
Nuevo Manual de la UNESCO para la Enseñanza de CienciasJames Smith
Este documento presenta el prefacio de una nueva edición del Manual de la Unesco para la Enseñanza de las Ciencias. Explica que la nueva edición actualiza el contenido y proporciona más material científico para cursos introductorios de ciencias. Detalla el proceso de revisión llevado a cabo por expertos de varios países bajo la coordinación de la Universidad de Maryland. El objetivo del manual es proveer ideas y recursos para que los maestros puedan enseñar ciencias de manera práctica utilizando materiales disponibles local
Calculating the Angle between Projections of Vectors via Geometric (Clifford)...James Smith
We express a problem from visual astronomy in terms of Geometric (Clifford) Algebra, then solve the problem by deriving expressions for the sine and cosine of the angle between projections of two vectors upon a plane. Geometric Algebra enables us to do so without deriving expressions for the projections themselves.
Estimation of the Earth's "Unperturbed" Perihelion from Times of Solstices an...James Smith
Published times of the Earth's perihelions do not refer to the perihelions of the orbit that the Earth would follow if unaffected by other bodies such as the Moon. To estimate the timing of that ``unperturbed" perihelion, we fit an unperturbed Kepler orbit to the timings of the year 2017's equinoxes and solstices. We find that the unperturbed 2017 perihelion, defined in that way, would occur 12.93 days after the December 2016 solstice. Using that result, calculated times of the year 2017's solstices and equinoxes differ from published values by less than five minutes. That degree of accuracy is sufficient for the intended use of the result.
Projection of a Vector upon a Plane from an Arbitrary Angle, via Geometric (C...James Smith
We show how to calculate the projection of a vector, from an arbitrary direction, upon a given plane whose orientation is characterized by its normal vector, and by a bivector to which the plane is parallel. The resulting solutions are tested by means of an interactive GeoGebra construction.
Formulas and Spreadsheets for Simple, Composite, and Complex Rotations of Vec...James Smith
We show how to express the representations of single, composite, and "rotated" rotations in GA terms that allow rotations to be calculated conveniently via spreadsheets. Worked examples include rotation of a single vector by a bivector angle; rotation of a vector about an axis; composite rotation of a vector; rotation of a bivector; and the "rotation of a rotation". Spreadsheets for doing the calculations are made available via live links.
How to Effect a Composite Rotation of a Vector via Geometric (Clifford) AlgebraJames Smith
We show how to express the representation of a composite rotation in terms that allow the rotation of a vector to be calculated conveniently via a spreadsheet that uses formulas developed, previously, for a single rotation. The work presented here (which includes a sample calculation) also shows how to determine the bivector angle that produces, in a single operation, the same rotation that is effected by the composite of two rotations.
A Modification of the Lifshitz-Slyozov-Wagner Equation for Predicting Coarsen...James Smith
The story behind this article is instructive, and even a bit troubling. I wrote it in 1991 as a continuation of part of my Doctoral thesis, which I’d completed a few years earlier. During that research, I’d found that scientists who’d done very fine laboratory work on Ostwald ripening during the 1960s had made a curious error in simple mass balances when deriving a rate equation for Ostwald ripening starting from the minimum-entropy-production-rate (MEPR) principle.
That error led the 1960s scientists to reject (with commendable honesty) their hypothesis that the MEPR principle is applicable to Ostwald ripening. Like all the rest of us metallurgists back then, I didn’t catch that error, until I examined the derivation of the MEPR-based rate equation in detail during my thesis work. However, I didn’t manage to re-derive the rate equation fully until I took up the subject again in the early 1990s. The scientists who did such fine lab work in the 1960s would no doubt have been pleased to learn that their empirical results agreed quite well with predictions made by the corrected equation. Thus, those scientists were correct in their hypothesis about the MEPR principle’s applicability.
I continue to wonder how we metallurgists overlooked, for more than two decades, the simple error that led those scientists to conclude, mistakenly but honestly, that they’d been wrong.
I never did manage to publish this article, but the same derivations and analyses were published by other researchers within a few years. Some of the reviewers’ comments on the article are addressed in the second article in this document, “Comments on ‘Ostwald Ripening Growth Rate for Nonideal Systems with Significant Mutual Solubility’”.
Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
Thinking of getting a dog? Be aware that breeds like Pit Bulls, Rottweilers, and German Shepherds can be loyal and dangerous. Proper training and socialization are crucial to preventing aggressive behaviors. Ensure safety by understanding their needs and always supervising interactions. Stay safe, and enjoy your furry friends!
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
Assessment and Planning in Educational technology.pptxKavitha Krishnan
In an education system, it is understood that assessment is only for the students, but on the other hand, the Assessment of teachers is also an important aspect of the education system that ensures teachers are providing high-quality instruction to students. The assessment process can be used to provide feedback and support for professional development, to inform decisions about teacher retention or promotion, or to evaluate teacher effectiveness for accountability purposes.
How to Calculate Distances from Centerline to Inside Walls of Domes, for Any Given Height above the Ground
1. 1
Calculation for Domes: Distance from Centerline to Inner Wall
for Any Given Height above Ground
This document describes the basis for the calculation, and provides examples.
Basis of the calculation
The vertical cross-sections of the domes considered here are circular arcs. The properties of circles allow us to
calculate the distance of interest via the Pythagorean Theorem.
Relevant properties of circles
All points on a circle are at the same distance from its center. That distance is called the radius of the circle.
Confusingly, the same word (radius) is also used for any segment from the center to a point along the circle:
Now, we’ll add to our diagram a horizontal line passing through the circle’s center. Please note that this horizontal
line represents the springline of the dome.
If we now draw a vertical segment, as shown below,
then a right triangle is formed by the vertical segment, the radius, and the segment from the center of the circle to
the point where the vertical segment intersects the springline:
Center
Radius
Center
Radius
Center
Radius
2. 2
Notice that the hypotenuse of our right triangle is a radius drawn to the point where the vertical segment touches
the circle.
Of course, we could also form a right triangle by first drawing a horizontal line at some height H above the
springline.
If the height H is small enough, then line we just drew will cut our circle at two points.
1
We’ll choose one of those
points, and call it P.
In a dome, point P would be on the inside of the wall, at height H above the springline:
Now, we’ll draw a vertical segment from P to the springline:
1
If H is equal to the radius of the circle, then the dashed line will be tangent to the circle. In other words, the
dashed line will cut the circle at only one point, which is at the top of the circle.
H
P
H
Center
Radius
3. 3
Next, we form the right angle by drawing a horizontal segment from the center of the circle to the point at which
our vertical segment intersects the springline. We’ve now formed our right angle:
Finally, we add the hypotenuse (the radius drawn to P).
Our right triangle is now complete.
We would follow the same procedure to construct the right triangle for any height that interests us. Here are some
examples:
The length of the hypotenuse is the same for all such triangles, since it’s equal to the radius of the circle. Notice
that we needn’t use the full circle to construct our right triangles. We could use only a semicircle,
H
P
H
P
H
P
4. 4
or even just the part of the circle that corresponds to the dome we wish to construct:
To continue, we’ll examine one such triangle in greater detail.
We’ve added the point Q, which is directly over the center of the circle, and at the same height as P. Note that the
length of the base of the triangle is equal to the distance from Q to P.
The angle at Q is a right angle,
so we now have two different right triangles, as shown below:
Both of the right triangles have the same hypotenuse (whose length is equal to the radius of the circle):Therefore,
the two red triangles (above) are identical. We can use either to calculate the distances we’ll need for constructing
a dome. Since we’re interested in finding distances from the axis of the dome to the inside of the wall, we’ll use
Triangle II in the sections that follow.
The Pythagorean Theorem
For convenience, we’ll re-draw Triangle II from the previous section, labeling the length of each side of the triangle
to help us apply the Pythagorean Theorem:
Q P Q P
I II
Q P
Q P
5. 5
According to the Pythagorean Theorem, the relationship between X, H, and R is
𝑋2
+ 𝐻2
= 𝑅2
,
from which
𝑋 = √𝑅2 − 𝐻2 .
Calculation of Distances from the Centerline of the Dome
to the Inside of the Wall, for Any Height above the Springline
In a hemispherical dome, the line L coincides with the centerline of the dome. Therefore, the distance from the
dome’s centerline to its inside wall, at any height H that interests us, is just X. In such a case, the equation we just
saw, above, is exactly the one we need:
In a hemispherical dome,
𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝑐𝑒𝑛𝑡𝑒𝑟𝑙𝑖𝑛𝑒 𝑡𝑜 𝑖𝑛𝑠𝑖𝑑𝑒 𝑤𝑎𝑙𝑙 𝑎𝑡 ℎ𝑒𝑖𝑔ℎ𝑡 𝐻 = 𝑋 = √𝑅2 − 𝐻2 .
RH
L
X
X
RH
R is the radius of the circle (and
therefore of the curve of the wall).
H is whatever height above the
springline that might interest us.
L Is the (imaginary) vertical line
that passes through the center of
the circle.
X is the distance that there is
between L and the inside wall of
the dome, at height H.
L
6. 6
Many domes for earthbag buildings are not hemispherical; instead, their vertical cross-section has the following
shape:
Each side is an arc of a circle whose center is offset from the centerline (CL) of the dome:
How can we find the distance between the centerline of the dome and the inside wall in this case? To simplify the
explanation, we’ll make use of the dome’s symmetry about its centerline. Because of that symmetry, we need
consider only one side:
From our previous work, we know how to find the distance X for any given height H:
L CL
R
CL
R
R
The radius R is the
same for both sides.
7. 7
But the distance we need to know isn’t X; instead, instead, it’s d, in the diagram below:
Examining this diagram, we see that if we know X, we can find d by subtracting the distance between the lines L
and CL,
d = X – (distance between lines L and CL),
or, since we know that 𝑿 = √𝑹 𝟐 − 𝑯 𝟐 ,
d = √𝑹 𝟐 − 𝑯 𝟐– (distance between lines L and CL).
Note that the distance between lines L and CL is also the distance between the center of the arc and the center of
the base of the dome. This observation allows us to put our answer in the following form:
L CL
R
X
H
d
L CL
R
X
H
𝑿 = √𝑹 𝟐 − 𝑯 𝟐
8. 8
Sample Calculations
We’ll do our sample calculation for a dome with the characteristics shown in the next figure.
The dome is symmetrical. Each side is an arc with a radius of 7.5m, and a center that’s offset 4.5 m from the
centerline of the dome:
6m
6m
To find d for any height H:
Step 1: Find X, from 𝑿 = √𝑹 𝟐 − 𝑯 𝟐.
Step 2: Find d from d = X - J.
L CL
R
X
H
d
J
(Distance between center
of circular arc and center
of base of dome.)
9. 9
Now, we’ll make three sample calculations. The first two will be done to validate our formulas, and our
understanding of them.
The first sample calculation: H = 0
If we study our first drawing,
We can see that at the springline, the distance between the centerline of the dome and the inside of the wall is
3m. Is this the result that we get from the procedure we developed earlier? We’ll now see.
6m
6m
L CL
R =7.5m
4.5m
10. 10
At the springline, H = 0. We’ll now follow our procedure,
Step 1: Find X, from 𝑿 = √𝑹 𝟐 − 𝑯 𝟐.
𝑿 = √𝟕. 𝟓 𝟐 − 𝟎 𝟐 = 7.5m.
Step 2: Find d from d = X - J.
d = 7.5 – 4.5 = 3m. OK
The Second Sample Calculation: H = 6m
If we study our first drawing again,
We’ll see that at a height of 6m above the springline, the distance between the inner wall and centerline is zero.
This observation allows us to make another check of our procedure.
6m
6m
To find d for any height H:
Step 1: Find X, from 𝑿 = √𝑹 𝟐 − 𝑯 𝟐.
Step 2: Find d from d = X - J.
L CL
R = 7.5m
X
H
d
J = 4.5m
(Distance between center
of circular arc and center
of base of dome.)
11. 11
Again, R = 7.5m, J = 4.5m, and H = 6m, so according to our procedure,
Step 1: Find X, from 𝑿 = √𝑹 𝟐 − 𝑯 𝟐.
𝑿 = √𝟕. 𝟓 𝟐 − 𝟔 𝟐 = 4.5m.
Step 2: Find d from d = X - J.
d = 4.5 – 4.5 = 0m. OK
The Third Sample Calculation: H = 3m
Now, we’ll calculate the distance from centerline to inner wall at 3m above the springline.
Again, R = 7.5m, J = 4.5m, but his time H = 3m, so according to our procedure,
Step 1: Find X, from 𝑿 = √𝑹 𝟐 − 𝑯 𝟐.
𝑿 = √𝟕. 𝟓 𝟐 − 𝟑 𝟐 = 6.874m.
Step 2: Find d from d = X - J.
d = 6.874 – 4.5 = 2.374m.