Sara Math Potw Solution
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Sara Dufresne outlines a solution to find the shaded area between two overlapping circles. She first calculates the area of each circle and triangle formed between them. She then finds the area of each sector formed by the triangles. Finally, she determines the shaded area by taking the difference between the total triangle area and summed sector areas.
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Sara Math Potw Solution
The document provides steps to solve a problem involving finding the shaded area between two overlapping circles. It first calculates the area of each circle as 16 square units. It then finds the area of an equilateral triangle formed between the circles as 8 square units. Finally, it calculates the area of three sectors formed by the triangles as 8/3 square units each and subtracts their total from the triangle area to get the shaded area of 4 square units.
Summary of trigonometric formulas2
This document summarizes trigonometric formulas for circles, right triangles, oblique triangles, and triangle areas. For circles, it provides formulas for arc length and sector area based on the radius and central angle. For right triangles, it covers the sine, cosine, and tangent definitions and the Pythagorean theorem. For oblique triangles, it introduces the law of cosines and law of sines to solve any triangle given sufficient information. It also provides three formulas for finding a triangle's area based on available side or angle measurements.
Trigonometry part 1
This math presentation tutorial provides a basic introduction to trigonometry. It explains trigonometry ratios, how to evaluate it using the right triangle trigonometry and SOHCAHTOA. In trigonometry explains hypotenuse, adjacent and opposite side.
Funciones Trigonometricas
The document defines trigonometric functions as the ratio of sides of a right triangle associated with its angles. It introduces the six basic trigonometric functions - sine, cosine, tangent, cosecant, secant and cotangent. It also discusses the Law of Cosines, which allows calculating the value of one side of a triangle given the angle opposite it and the other two sides. The Law of Sines relates the ratios of sides to sines of angles in any triangle. Finally, it presents several trigonometric identities, such as reciprocal, quotient and Pythagorean identities, that are true for all values of the variables involved.
Trigonometry 1
Trigonometry deals with measurements of sides and angles of triangles and their relationships. For right triangles, there are three main trigonometric ratios: sine, cosine, and tangent. These ratios relate the lengths of sides of a right triangle to an angle of the triangle. The document provides examples of calculating trigonometric ratios using SOHCAHTOA and finding unknown angles in a right triangle given side lengths.
4.11.3 Similar Right Triangles
The student can use the geometric mean to find segment lengths in right triangles and apply similarity relationships to solve problems. Specifically:
- The altitude to the hypotenuse of a right triangle forms two triangles similar to each other and the original triangle. Proportions can be set up between corresponding sides.
- The length of the altitude to the hypotenuse is the geometric mean of the lengths of the two segments it forms.
- The length of a leg is the geometric mean of the lengths of the hypotenuse and the adjacent segment. Setting up similar triangles allows solving for unknown lengths.
Perimeter
The document defines perimeter as the total length of the sides surrounding a shape. It provides the formulas to calculate the perimeter of common shapes like triangles, squares, rectangles, pentagons, and circles. For each shape, it gives an example with measurements and calculates the perimeter. The perimeter of a triangle is the sum of the lengths of its three sides. The perimeter of a square is the sum of the lengths of its four equal sides. The perimeter of a rectangle is calculated by adding the lengths of all four of its sides.
Trigonomerty
This document discusses how to use trigonometric ratios (SOH CAH TOA) to solve for unknown sides of a right triangle. It explains that SOH relates sine to the opposite and hypotenuse sides, CAH relates cosine to the adjacent and hypotenuse sides, and TOA relates tangent to the opposite and adjacent sides. The document provides step-by-step instructions: 1) name the sides of the triangle based on the given angle, 2) use SOH, CAH or TOA to identify the appropriate ratio, 3) write the ratio as a fraction and plug in known values, 4) use a calculator to solve for the last side.
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Sara Math Potw Solution
The document provides steps to solve a problem involving finding the shaded area between two overlapping circles. It first calculates the area of each circle as 16 square units. It then finds the area of an equilateral triangle formed between the circles as 8 square units. Finally, it calculates the area of three sectors formed by the triangles as 8/3 square units each and subtracts their total from the triangle area to get the shaded area of 4 square units.
Summary of trigonometric formulas2
This document summarizes trigonometric formulas for circles, right triangles, oblique triangles, and triangle areas. For circles, it provides formulas for arc length and sector area based on the radius and central angle. For right triangles, it covers the sine, cosine, and tangent definitions and the Pythagorean theorem. For oblique triangles, it introduces the law of cosines and law of sines to solve any triangle given sufficient information. It also provides three formulas for finding a triangle's area based on available side or angle measurements.
Trigonometry part 1
This math presentation tutorial provides a basic introduction to trigonometry. It explains trigonometry ratios, how to evaluate it using the right triangle trigonometry and SOHCAHTOA. In trigonometry explains hypotenuse, adjacent and opposite side.
Funciones Trigonometricas
The document defines trigonometric functions as the ratio of sides of a right triangle associated with its angles. It introduces the six basic trigonometric functions - sine, cosine, tangent, cosecant, secant and cotangent. It also discusses the Law of Cosines, which allows calculating the value of one side of a triangle given the angle opposite it and the other two sides. The Law of Sines relates the ratios of sides to sines of angles in any triangle. Finally, it presents several trigonometric identities, such as reciprocal, quotient and Pythagorean identities, that are true for all values of the variables involved.
Trigonometry 1
Trigonometry deals with measurements of sides and angles of triangles and their relationships. For right triangles, there are three main trigonometric ratios: sine, cosine, and tangent. These ratios relate the lengths of sides of a right triangle to an angle of the triangle. The document provides examples of calculating trigonometric ratios using SOHCAHTOA and finding unknown angles in a right triangle given side lengths.
4.11.3 Similar Right Triangles
The student can use the geometric mean to find segment lengths in right triangles and apply similarity relationships to solve problems. Specifically:
- The altitude to the hypotenuse of a right triangle forms two triangles similar to each other and the original triangle. Proportions can be set up between corresponding sides.
- The length of the altitude to the hypotenuse is the geometric mean of the lengths of the two segments it forms.
- The length of a leg is the geometric mean of the lengths of the hypotenuse and the adjacent segment. Setting up similar triangles allows solving for unknown lengths.
Perimeter
The document defines perimeter as the total length of the sides surrounding a shape. It provides the formulas to calculate the perimeter of common shapes like triangles, squares, rectangles, pentagons, and circles. For each shape, it gives an example with measurements and calculates the perimeter. The perimeter of a triangle is the sum of the lengths of its three sides. The perimeter of a square is the sum of the lengths of its four equal sides. The perimeter of a rectangle is calculated by adding the lengths of all four of its sides.
Trigonomerty
This document discusses how to use trigonometric ratios (SOH CAH TOA) to solve for unknown sides of a right triangle. It explains that SOH relates sine to the opposite and hypotenuse sides, CAH relates cosine to the adjacent and hypotenuse sides, and TOA relates tangent to the opposite and adjacent sides. The document provides step-by-step instructions: 1) name the sides of the triangle based on the given angle, 2) use SOH, CAH or TOA to identify the appropriate ratio, 3) write the ratio as a fraction and plug in known values, 4) use a calculator to solve for the last side.
Evidence for Pi
This document discusses geometric relationships between triangles inscribed in circles. It introduces the concept of special right triangles formed at an angle it calls pi, which allows triangles to be joined without gaps. This pi angle is based on the golden ratio phi and allows the geometry to have perfect symmetry and balance where terms are reciprocals. The document claims this proves pi can be written as a rational number and visualized as a fractal, challenging the conventional view of pi.
Core 3 Simpsons Rule
Thomas Simpson published Simpson's Rule in 1743, which uses parabolas instead of straight lines to better approximate the area under a curve. Simpson's Rule requires an even number of intervals or ordinates and is more accurate than the Trapezium Rule. Two examples are provided where Simpson's Rule is used with either five ordinates or four intervals to estimate the value of integrals.
Inequality theorem (l4)
The triangle inequality theorem states that the sum of any two side lengths of a triangle must be greater than the third side length. This means that if you are given two side lengths, you can determine the possible range of values for the third side length. The theorem can also be used to determine if three given side lengths could form a triangle. An example problem shows that sides of lengths 1, 2, and 3 do not form a triangle because 1 + 2 is not greater than 3.
Right triangle trigonometry
This document provides an overview of trigonometric functions and identities. It defines the basic trig functions of sine, cosine, and tangent using ratios of sides in a right triangle. It introduces reciprocal functions and shows how to derive trig identities from fundamental relationships like the Pythagorean theorem. Key identities covered include reciprocal, quotient, and Pythagorean identities. Methods are presented for finding all trig functions of an acute angle given one function value through using definitions, identities, or converting to a complementary angle.
Trigonometry
Trigonometry is a branch of mathematics that deals with relationships between angles and sides of right triangles. It originated in ancient Greece. There are three basic trigonometric ratios - sine, cosine, and tangent - that relate the measures of angles to the lengths of sides in a right triangle. To solve problems using trigonometric ratios, one identifies the appropriate ratio based on the given and unknown sides, then substitutes the known values into the ratio formula and solves for the unknown.
4.11.4 Trigonometry
Define the sine, cosine, and tangent ratios and their inverses
Find the measure of a side given a side and an angle
Find the measure of an angle given two sides
Use trig ratios to solve problems
Trigonometric ratios
This document introduces the SOH CAH TOA mnemonic for remembering the definitions of the trigonometric ratios sine, cosine, and tangent. It states that SOH CAH TOA is a trigonometric ratio and defines sine as opposite over hypotenuse, cosine as adjacent over hypotenuse, and tangent as opposite over adjacent. It then provides examples of using trigonometric ratios to solve for unknown angles and sides of right triangles.
4.11.5 Solving Right Triangles
This document provides instructions for solving right triangles. It explains that to solve a right triangle, you find all the missing parts. If given an angle and side, you can use trig ratios to find the other acute angle and missing sides. If given two sides, you can use the Pythagorean theorem to find the missing side and trig ratios to find an angle. It provides examples of solving right triangles when given an angle and side or two sides.
Perimeter
The document defines perimeter and provides formulas for calculating the perimeter of rectangles, squares, and triangles. It explains that perimeter is the distance around a two-dimensional shape. For rectangles, the perimeter formula is P=2(length+breadth). For squares, the formula is P=4xside. For triangles, the formula is P=a+b+c, where a, b, and c are the lengths of the three sides. An example problem calculates the perimeter of a triangular plot of land that is to be fenced with four rounds of wire.
Lesson 5 4
The document discusses triangle inequalities and related concepts:
1) The smallest angle is opposite the shortest side and largest angle is opposite longest side.
2) The sum of any two sides of a triangle is greater than the third side.
3) The range of a third side of a triangle given two other sides is from the difference to the sum of the two known sides.
Right triangles
This document provides instruction on solving right triangles using trigonometric ratios. It includes examples of identifying angles from trigonometric ratios, calculating angle measures from ratios using inverse functions, solving right triangles when given side lengths or an angle measure, solving triangles in the coordinate plane, and applying trigonometry to problems involving road grades. Students are guided through applying concepts to multiple examples and practice problems.
Obj. 40 Trigonometry
For any right triangle
Define the sine, cosine, and tangent ratios and their inverses
Find the measure of a side given a side and an angle
Find the measure of an angle given two sides
Use trig ratios to solve problems
Right triangle trigonometry
Trigonometric ratios are based on similar right triangles and can be used to relate the sides of a right triangle to an angle. There are six trigonometric ratios: sine, cosine, tangent, cotangent, secant, and cosecant. These ratios use the hypotenuse, opposite, and adjacent sides of a right triangle and can be remembered using the mnemonic SOHCAHTOA. The ratios allow for solving for unknown sides or angles of a right triangle when sufficient information is provided.
Examples Of Perimeter
The document provides 3 examples of perimeter calculations: 1) Finding the perimeter of a triangle with sides of 5, 9, and 11 cm, which equals 25 cm. 2) Calculating the perimeter of a rectangle and two similar squares shown in a diagram, which equals 34 cm. 3) Determining the cost of barbed wire to fence a 120m x 120m square garden, which is RM5280 using 480m of fencing at RM11 per meter.
similar triangles
1) Triangles are three-sided polygons formed by three line segments. The sum of the three interior angles is always 180 degrees.
2) There are certain properties that apply to all triangles, such as being rigid flat shapes that satisfy the Triangle Inequality.
3) Triangles can be categorized based on their sides and angles. Congruent triangles are identical in shape and size, while similar triangles have the same shape but may differ in size. The properties of similar triangles can be used to solve proportional relationships.
simpion's 3/8 rule
The document is a presentation on Simpson's 3/8 rule. It introduces numerical integration and Simpson's rule as an improvement over the midpoint and trapezium rules. It explains that Simpson's rule has two variations: Simpson's 1/3 rule and Simpson's 3/8 rule. Simpson's 3/8 rule is based on a cubic interpolation and is also known as Simpson's 2nd rule, using a polynomial of degree 3. It provides an example C program for implementing Simpson's 3/8 rule when the number of strips is a multiple of three.
Perimeter and Area of Polygons
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. Area is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object
5 2 triangle inequality theorem
The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. For a triangle to exist, the sum of the lengths of two sides must be greater than the length of the third side. The possible length of the third side of a triangle with sides of 6 and 12 must be greater than 6 and less than 18.
Lesson 12: Right Triangle Trigonometry
This document contains a lesson on right triangle trigonometry from Lehman College's Department of Mathematics. It includes definitions of the tangent, sine and cosine ratios, examples of solving right triangles using trigonometric ratios, and a derivation of the Pythagorean identity that relates sine and cosine functions. Key concepts covered are the definitions of trigonometric ratios in right triangles, using trigonometric functions to solve for unknown side lengths, and deriving trigonometric identities from the Pythagorean theorem.
Inequalities in a triangle
This document contains instructions and examples for identifying properties of triangles based on side lengths and angle measures. It begins by asking the student to draw a triangle with given side lengths and measure its angles. Several practice problems then ask the student to name the largest and smallest angles of triangles where the side lengths are given. Another set of problems has the student arrange the sides of triangles in order of increasing length. The document concludes by asking the student to identify relationships between side lengths and angle measures using inequality symbols.
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This document discusses geometric relationships between triangles inscribed in circles. It introduces the concept of special right triangles formed at an angle it calls pi, which allows triangles to be joined without gaps. This pi angle is based on the golden ratio phi and allows the geometry to have perfect symmetry and balance where terms are reciprocals. The document claims this proves pi can be written as a rational number and visualized as a fractal, challenging the conventional view of pi.
Core 3 Simpsons Rule
Thomas Simpson published Simpson's Rule in 1743, which uses parabolas instead of straight lines to better approximate the area under a curve. Simpson's Rule requires an even number of intervals or ordinates and is more accurate than the Trapezium Rule. Two examples are provided where Simpson's Rule is used with either five ordinates or four intervals to estimate the value of integrals.
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The triangle inequality theorem states that the sum of any two side lengths of a triangle must be greater than the third side length. This means that if you are given two side lengths, you can determine the possible range of values for the third side length. The theorem can also be used to determine if three given side lengths could form a triangle. An example problem shows that sides of lengths 1, 2, and 3 do not form a triangle because 1 + 2 is not greater than 3.
Right triangle trigonometry
This document provides an overview of trigonometric functions and identities. It defines the basic trig functions of sine, cosine, and tangent using ratios of sides in a right triangle. It introduces reciprocal functions and shows how to derive trig identities from fundamental relationships like the Pythagorean theorem. Key identities covered include reciprocal, quotient, and Pythagorean identities. Methods are presented for finding all trig functions of an acute angle given one function value through using definitions, identities, or converting to a complementary angle.
Trigonometry
Trigonometry is a branch of mathematics that deals with relationships between angles and sides of right triangles. It originated in ancient Greece. There are three basic trigonometric ratios - sine, cosine, and tangent - that relate the measures of angles to the lengths of sides in a right triangle. To solve problems using trigonometric ratios, one identifies the appropriate ratio based on the given and unknown sides, then substitutes the known values into the ratio formula and solves for the unknown.
4.11.4 Trigonometry
Define the sine, cosine, and tangent ratios and their inverses
Find the measure of a side given a side and an angle
Find the measure of an angle given two sides
Use trig ratios to solve problems
Trigonometric ratios
This document introduces the SOH CAH TOA mnemonic for remembering the definitions of the trigonometric ratios sine, cosine, and tangent. It states that SOH CAH TOA is a trigonometric ratio and defines sine as opposite over hypotenuse, cosine as adjacent over hypotenuse, and tangent as opposite over adjacent. It then provides examples of using trigonometric ratios to solve for unknown angles and sides of right triangles.
4.11.5 Solving Right Triangles
This document provides instructions for solving right triangles. It explains that to solve a right triangle, you find all the missing parts. If given an angle and side, you can use trig ratios to find the other acute angle and missing sides. If given two sides, you can use the Pythagorean theorem to find the missing side and trig ratios to find an angle. It provides examples of solving right triangles when given an angle and side or two sides.
Perimeter
The document defines perimeter and provides formulas for calculating the perimeter of rectangles, squares, and triangles. It explains that perimeter is the distance around a two-dimensional shape. For rectangles, the perimeter formula is P=2(length+breadth). For squares, the formula is P=4xside. For triangles, the formula is P=a+b+c, where a, b, and c are the lengths of the three sides. An example problem calculates the perimeter of a triangular plot of land that is to be fenced with four rounds of wire.
Lesson 5 4
The document discusses triangle inequalities and related concepts:
1) The smallest angle is opposite the shortest side and largest angle is opposite longest side.
2) The sum of any two sides of a triangle is greater than the third side.
3) The range of a third side of a triangle given two other sides is from the difference to the sum of the two known sides.
Right triangles
This document provides instruction on solving right triangles using trigonometric ratios. It includes examples of identifying angles from trigonometric ratios, calculating angle measures from ratios using inverse functions, solving right triangles when given side lengths or an angle measure, solving triangles in the coordinate plane, and applying trigonometry to problems involving road grades. Students are guided through applying concepts to multiple examples and practice problems.
Obj. 40 Trigonometry
For any right triangle
Define the sine, cosine, and tangent ratios and their inverses
Find the measure of a side given a side and an angle
Find the measure of an angle given two sides
Use trig ratios to solve problems
Right triangle trigonometry
Trigonometric ratios are based on similar right triangles and can be used to relate the sides of a right triangle to an angle. There are six trigonometric ratios: sine, cosine, tangent, cotangent, secant, and cosecant. These ratios use the hypotenuse, opposite, and adjacent sides of a right triangle and can be remembered using the mnemonic SOHCAHTOA. The ratios allow for solving for unknown sides or angles of a right triangle when sufficient information is provided.
Examples Of Perimeter
The document provides 3 examples of perimeter calculations: 1) Finding the perimeter of a triangle with sides of 5, 9, and 11 cm, which equals 25 cm. 2) Calculating the perimeter of a rectangle and two similar squares shown in a diagram, which equals 34 cm. 3) Determining the cost of barbed wire to fence a 120m x 120m square garden, which is RM5280 using 480m of fencing at RM11 per meter.
similar triangles
1) Triangles are three-sided polygons formed by three line segments. The sum of the three interior angles is always 180 degrees.
2) There are certain properties that apply to all triangles, such as being rigid flat shapes that satisfy the Triangle Inequality.
3) Triangles can be categorized based on their sides and angles. Congruent triangles are identical in shape and size, while similar triangles have the same shape but may differ in size. The properties of similar triangles can be used to solve proportional relationships.
simpion's 3/8 rule
The document is a presentation on Simpson's 3/8 rule. It introduces numerical integration and Simpson's rule as an improvement over the midpoint and trapezium rules. It explains that Simpson's rule has two variations: Simpson's 1/3 rule and Simpson's 3/8 rule. Simpson's 3/8 rule is based on a cubic interpolation and is also known as Simpson's 2nd rule, using a polynomial of degree 3. It provides an example C program for implementing Simpson's 3/8 rule when the number of strips is a multiple of three.
Perimeter and Area of Polygons
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. Area is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object
5 2 triangle inequality theorem
The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. For a triangle to exist, the sum of the lengths of two sides must be greater than the length of the third side. The possible length of the third side of a triangle with sides of 6 and 12 must be greater than 6 and less than 18.
Lesson 12: Right Triangle Trigonometry
This document contains a lesson on right triangle trigonometry from Lehman College's Department of Mathematics. It includes definitions of the tangent, sine and cosine ratios, examples of solving right triangles using trigonometric ratios, and a derivation of the Pythagorean identity that relates sine and cosine functions. Key concepts covered are the definitions of trigonometric ratios in right triangles, using trigonometric functions to solve for unknown side lengths, and deriving trigonometric identities from the Pythagorean theorem.
Inequalities in a triangle
This document contains instructions and examples for identifying properties of triangles based on side lengths and angle measures. It begins by asking the student to draw a triangle with given side lengths and measure its angles. Several practice problems then ask the student to name the largest and smallest angles of triangles where the side lengths are given. Another set of problems has the student arrange the sides of triangles in order of increasing length. The document concludes by asking the student to identify relationships between side lengths and angle measures using inequality symbols.
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إضغ بين إيديكم من أقوى الملازم التي صممتها
ملزمة تشريح الجهاز الهيكلي (نظري 3)
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تتميز هذهِ الملزمة بعِدة مُميزات :
1- مُترجمة ترجمة تُناسب جميع المستويات
2- تحتوي على 78 رسم توضيحي لكل كلمة موجودة بالملزمة (لكل كلمة !!!!)
#فهم_ماكو_درخ
3- دقة الكتابة والصور عالية جداً جداً جداً
4- هُنالك بعض المعلومات تم توضيحها بشكل تفصيلي جداً (تُعتبر لدى الطالب أو الطالبة بإنها معلومات مُبهمة ومع ذلك تم توضيح هذهِ المعلومات المُبهمة بشكل تفصيلي جداً
5- الملزمة تشرح نفسها ب نفسها بس تكلك تعال اقراني
6- تحتوي الملزمة في اول سلايد على خارطة تتضمن جميع تفرُعات معلومات الجهاز الهيكلي المذكورة في هذهِ الملزمة
واخيراً هذهِ الملزمة حلالٌ عليكم وإتمنى منكم إن تدعولي بالخير والصحة والعافية فقط
كل التوفيق زملائي وزميلاتي ، زميلكم محمد الذهبي 💊💊
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Internet resources for natural science
How to Download & Install Module From the Odoo App Store in Odoo 17
Custom modules offer the flexibility to extend Odoo's capabilities, address unique requirements, and optimize workflows to align seamlessly with your organization's processes. By leveraging custom modules, businesses can unlock greater efficiency, productivity, and innovation, empowering them to stay competitive in today's dynamic market landscape. In this tutorial, we'll guide you step by step on how to easily download and install modules from the Odoo App Store.
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In this slide we will discuss Creation or Updation of a mandatory field is not set in Odoo 17.
NIPER 2024 MEMORY BASED QUESTIONS.ANSWERS TO NIPER 2024 QUESTIONS.NIPER JEE 2...
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HYPERTENSION - SLIDE SHARE PRESENTATION.
IT WILL BE HELPFULL FOR THE NUSING STUDENTS
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220711130083 SUBHASHREE RAKSHIT Internet resources for social science
Internet resources for social science
A Visual Guide to 1 Samuel | A Tale of Two Hearts
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Slides from a Capitol Technology University webinar held June 20, 2024. The webinar featured Dr. Donovan Wright, presenting on the Department of Defense Digital Transformation.
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Sara Math Potw Solution
- 1. Sara Dufresne’s POTW Solution 11/30/09
- 2. First you want to find the area of each circle so you use the formula A= . So you get which equals . Next you want to find the area of each triangle so you use A= 1/2bh. The triangle you can form is equilateral with sides of 8. So you get ½(8)(4 ) which equals
- 3. Next you want to find the area of each sector that is formed by the triangle so you get 60 degrees/360 degrees x 16pie which can be reduced to 2degrees/3 degrees x 4pie which equals 8pie / 3. So the shaded area in between the circles is the difference between the area of the triangle and the sum of the three sectors. So you get 16root 3 – 3(8pie/30 which equals 16root 3 – 8pie.
- 4. The final answer is 16 root 3 – 8 square units.