SlideShare a Scribd company logo
Capstone Project
Science, Technology, Engineering, and Mathematics
Precalculus
Science, Technology, Engineering, and Mathematics
Lesson 4.3
Applications of
Hyperbolas in Real-life
Situations
2
Have you every
wondered how we
were able to track the
locations of ships and
aircrafts before we
had navigation
satellites (e.g. Global
Positioning System or
GPS)?
3
Navigation systems
such as the LORAN
(long-range
navigation) use radio
signals to determine
the exact location of a
ship or aircraft.
4
Knowledge of hyperbolas is used in this type of navigation
system.
5
What are some real-life
applications of hyperbolas?
Learning Competency
At the end of the lesson, you should be able to do the following:
6
Solve situational problems involving
hyperbolas (STEM_PC11AG-Ie-2).
Learning Objectives
At the end of the lesson, you should be able to do the following:
7
● Recall the different parts and properties of a
hyperbola.
● Solve word problems on hyperbolas.
8
A hyperbola is defined as a set of points on a plane
whose absolute difference between the distances from
the foci, 𝐹1 and 𝐹2, is constant.
Hyperbolas
9
Parts and Properties of Hyperbolas
Given arbitrary points 𝑃1 and 𝑃2 on the hyperbola,
|𝑃1𝐹1 − 𝑃1𝐹2| = 𝑃2𝐹1 − 𝑃2𝐹2 = 2𝑎.
What is the distance 𝑎?
10
Parts and Properties of Hyperbolas
𝒄2 = 𝒂2 + 𝒃2
11
How do you determine the value
of 𝒂 given the values of 𝒃 and 𝒄?
How do you determine the value
of 𝒃 given the values of 𝒂 and 𝒄?
12
Standard Equations of a Hyperbola
Orientation of
Principal Axis
Equation
Horizontal
𝒙 − 𝒉 𝟐
𝒂𝟐
−
𝒚 − 𝒌 𝟐
𝒃𝟐
= 𝟏
Vertical
𝒚 − 𝒌 𝟐
𝒂𝟐
−
𝒙 − 𝒉 𝟐
𝒃𝟐
= 𝟏
13
Applications of Hyperbolas
Hyperbolic
Navigation
Example: LORAN-C
(long-range navigation)
14
Applications of Hyperbolas
Hyperbolic
Navigation
Two stations, 𝐴 and 𝐵,
transmit radio signals.
The radio signals reach
the ship at different
times.
15
Applications of Hyperbolas
Hyperbolic
Navigation
The location 𝑃 of the
ship is somewhere on
a hyperbola.
16
Applications of Hyperbolas
Hyperbolic
Navigation
The difference
between the distances
of the two stations
from the ship at point
𝑃 is 𝑷𝑨 − 𝑷𝑩 = 𝟐𝒂.
17
Applications of Hyperbolas
Cooling Towers
Cooling towers use
water from rivers and
lakes to cool nuclear
power plants.
18
Applications of Hyperbolas
Cooling Towers
They protect aquatic
life by ensuring that
the water is returned
to the environment at
normal temperatures.
19
Applications of Hyperbolas
Cooling Towers
Cooling towers are
hyperboloid in shape.
Why?
20
Applications of Hyperbolas
Cooling Towers
 able to withstand
strong winds
 efficient in cooling
 economical
Let’s Practice!
21
Point 𝑷 is on a hyperbola. The distance of 𝑷 from the
first focus is 6 units more than its distance from the
second focus. What is the distance of the center to a
vertex of the hyperbola?
Let’s Practice!
22
3 units
Point 𝑷 is on a hyperbola. The distance of 𝑷 from the
first focus is 6 units more than its distance from the
second focus. What is the distance of the center to a
vertex of the hyperbola?
Try It!
23
23
Point 𝑷 is on a hyperbola. The
difference between the distances of 𝑷
from the first focus and the second
focus is 20 units. What is the distance
of the center to a vertex of the
hyperbola?
Let’s Practice!
24
What is the equation of a hyperbola whose center is
at the origin, has a horizontal principal axis, the
value of 𝒂 is 𝟒, and the point (𝟗, 𝟏𝟒) is on the
hyperbola?
Let’s Practice!
25
𝒙𝟐
𝟏𝟔
−
𝒚𝟐
𝟒𝟖. 𝟐𝟓
= 𝟏
What is the equation of a hyperbola whose center is
at the origin, has a horizontal principal axis, the
value of 𝒂 is 𝟒, and the point (𝟗, 𝟏𝟒) is on the
hyperbola?
Try It!
26
26
What is the equation of a hyperbola
whose center is at the origin, has a
vertical principal axis, the value of 𝒂
is 𝟖, and the point (𝟖, 𝟏𝟑) is on the
hyperbola?
Let’s Practice!
27
Point 𝑷 is on a hyperbola with a vertical principal
axis and whose center is at the origin. The distance
of 𝑷 from the first focus 𝑭𝟏 is 12 units more than its
distance from the second focus 𝑭𝟐. The distance
between the two foci is 20 units. What is the
equation of the hyperbola?
Let’s Practice!
28
𝒚𝟐
𝟑𝟔
−
𝒙𝟐
𝟔𝟒
= 𝟏
Point 𝑷 is on a hyperbola with a vertical principal
axis and whose center is at the origin. The distance
of 𝑷 from the first focus 𝑭𝟏 is 12 units more than its
distance from the second focus 𝑭𝟐. The distance
between the two foci is 20 units. What is the
equation of the hyperbola?
Try It!
29
29
Point 𝑷 is on a hyperbola with a
vertical principal axis and whose
center is at the origin. The distance of
𝑷 from the first focus 𝑭𝟏 is 10 units
more than its distance from the
second focus 𝑭𝟐. The distance between
the two foci is 26 units. What is the
equation of the hyperbola?
Let’s Practice!
30
The sides of a cooling tower represent a hyperbola. The
width of the slimmest part of the cooling tower is 50
meters. The length from this point to the top is 64
meters. The diameter of the top of the cooling tower is
60 meters. Find the equation of the hyperbola that
represents the sides of the cooling tower. Set the
middle of the slimmest part as the origin.
Let’s Practice!
31
𝒙𝟐
𝟔𝟐𝟓
−
𝒚𝟐
𝟗𝟑𝟎𝟗. 𝟎𝟗
= 𝟏
The sides of a cooling tower represent a hyperbola. The
width of the slimmest part of the cooling tower is 50
meters. The length from this point to the top is 64
meters. The diameter of the top of the cooling tower is
60 meters. Find the equation of the hyperbola that
represents the sides of the cooling tower. Set the
middle of the slimmest part as the origin.
Try It!
32
32
The sides of a cooling tower represent a
hyperbola. The width of the slimmest part
of the cooling tower is 54 meters. The
length from this point to the top is 70
meters. The diameter of the top of the
cooling tower is 68 meters. Find the
equation of the hyperbola that represents
the sides of the cooling tower. Set the
middle of the slimmest part as the origin.
Let’s Practice!
33
Two stations 𝑨 and 𝑩 are along a straight coast such
that station 𝑨 is 100 miles west of station 𝑩. They both
transmit radio signals at the speed of 186 000 miles per
second. A ship sailing at sea is 50 miles from the coast.
The radio signal from station 𝑩 arrives at the ship
0.0002 of a second earlier than the radio signal sent
from station 𝑨. Where is the ship? Set the midpoint of 𝑨
and 𝑩 as the origin.
Let’s Practice!
34
(𝟐𝟕. 𝟑𝟒, 𝟓𝟎)
Two stations 𝑨 and 𝑩 are along a straight coast such
that station 𝑨 is 100 miles west of station 𝑩. They both
transmit radio signals at the speed of 186 000 miles per
second. A ship sailing at sea is 50 miles from the coast.
The radio signal from station 𝑩 arrives at the ship
0.0002 of a second earlier than the radio signal sent
from station 𝑨. Where is the ship? Set the midpoint of 𝑨
and 𝑩 as the origin.
Try It!
35
35
Two stations 𝑨 and 𝑩 transmit radio signals
such that station A is 200 miles west of station
𝑩. Both stations sent radio signals with a speed
of 𝟎. 𝟐 miles per microsecond to a plane
traveling west. The signal from station 𝑩
reaches a plane 500 microseconds faster than
the signal from station 𝑨. If the plane is 80
miles north of the line from stations 𝑨 to 𝑩,
what is the location of the plane? Set the
midpoint of 𝑨 and 𝑩 as the origin.
Check Your Understanding
36
Let 𝑷 be a point on a hyperbola with center at the
origin and foci 𝑭𝟏 and 𝑭𝟐. Answer the following
questions. Round off your answer to two decimal
places.
1. If 𝑃𝐹1 − 𝑃𝐹2 = 18, what is 𝑎?
2. If 𝑃𝐹2 − 𝑃𝐹1 = 32 and 𝑏 = 12, what is 𝑐?
3. If 𝑃𝐹1 − 𝑃𝐹2 = 34 and 𝑏2 = 256, what is 𝑐?
4. If 𝑐 = 39 and 𝑎 = 36, what is 𝑏?
5. If 𝑐2 = 1 600 and 𝑎 = 32, what is 𝑏?
Check Your Understanding
37
Solve the following problems.
1. A nuclear power plant has a cooling tower whose sides
are in the shape of a hyperbola. The slimmest part of
the tower is 58 meters. The length from this point to
the top is 80 meters. The radius of the top of the
cooling tower is 72 meters. What is the equation of the
hyperbola that represents the sides of the cooling
tower? Set the middle of the slimmest part as the
origin.
Check Your Understanding
38
Solve the following problems.
2. The sides of a cooling tower represent a hyperbola. The
width of the slimmest part of the cooling tower is 80
meters. The slimmest part of the tower is 90 meters
from the ground. The diameter of the base of the tower
is 130 meters. What is the equation of the hyperbola
that represents the sides of the cooling tower? Set the
middle of the slimmest part as the origin.
Check Your Understanding
39
Solve the following problems.
3. Two radio signaling stations 𝐴 and 𝐵 are 120 kilometers
apart. The radio signal from the two stations has a
speed of 300 000 kilometers per second. A ship at sea
receives the signals such that the signal from station 𝐵
arrives 0.0002 seconds before the signal from station 𝐴.
What is the equation of the hyperbola where the ship is
located? Set the midpoint of 𝐴 and 𝐵 as the origin.
Let’s Sum It Up!
40
● A hyperbola is a set of points on a plane whose
absolute difference between the distances from two
fixed points 𝐹1 and 𝐹2 is constant.
● Given any point 𝑃 on a hyperbola with foci 𝐹1 and 𝐹2,
𝑃𝐹1 − 𝑃𝐹2 = 2𝑎.
Let’s Sum It Up!
41
● The distance from the center to a vertex of the
hyperbola is 𝒂, the distance from the center to an
endpoint of the conjugate axis is 𝒃, and the focal
distance is 𝒄.
● The relationship between the three distances is
𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐.
Let’s Sum It Up!
42
● The standard forms of equation of a hyperbola whose
center is at the origin are
𝒙𝟐
𝒂𝟐 −
𝒚𝟐
𝒃𝟐 = 𝟏 if the principal axis
is horizontal, and
𝒚𝟐
𝒂𝟐 −
𝒙𝟐
𝒃𝟐 = 𝟏 if the principal axis is
vertical.
Let’s Sum It Up!
43
● In the real world, hyperbolas are used for navigation
and the structure of cooling towers.
Let’s Sum It Up!
44
● To solve word problems on hyperbolas, follow these
steps:
1. Determine the standard form of equation of the
hyperbola.
2. Draw an illustration to be able to visualize the
problem.
3. Solve for the unknown equation or values.
Key Formulas
45
Concept Formula Description
Relationship between
the values 𝒂, 𝒃, and 𝒄
𝑎2 + 𝑏2 = 𝑐2,
where 𝑎 is the distance
from the center to a
vertex, 𝑏 is the distance
from the center to an
endpoint of the
conjugate axis, and 𝑐 is
the focal distance.
Use this formula to find
an unknown distance
(e.g., focal distance)
when the other values
are known.
Key Formulas
46
Concept Formula Description
Equation of a
Hyperbola in Standard
Form
𝑥 − ℎ 2
𝑎2 −
𝑦 − 𝑘 2
𝑏2 = 1,
where
 (ℎ, 𝑘) is the center;
 𝑎 is the distance from
the center to a
vertex, and
 𝑏 is the distance from
the center to an
endpoint of the
conjugate axis.
Use this formula to find
the equation of a
hyperbola given its
center, 𝑎, and 𝑏 if the
transverse axis is
horizontal.
Key Formulas
47
Concept Formula Description
Equation of a
Hyperbola in Standard
Form
𝑦 − 𝑘 2
𝑎2 −
𝑥 − ℎ 2
𝑏2 = 1,
where
 (ℎ, 𝑘) is the center;
 𝑎 is the distance from
the center to a
vertex, and
 𝑏 is the distance from
the center to an
endpoint of the
conjugate axis.
Use this formula to find
the equation of a
hyperbola given its
center, 𝑎, and 𝑏 if the
transverse axis is
vertical.
Challenge Yourself
48
48
A nuclear power plant has a cooling tower
that is hyperboloid in shape. The width of
the slimmest part of the cooling tower is 60
meters and is 85 meters from the ground.
The diameter of the base of the tower is 120
meters, while the diameter of the top is 96
meters. What is the height of the tower?
Photo Credits
49
● Slides 2 and 3: Navigation boat Engineer Matusevich by Torin is licensed under CC BY-SA 3.0 via
Wikimedia Commons.
● Slide 13: Loran C Navigator by Morn the Gorn is licensed under CC BY-SA 3.0 via Wikimedia
Commons.
● Slides 17 to 20: Cooling towers of Dukovany Nuclear Power Plant in Dukovany, Třebíč District by
Jiří Sedláček is licensed under CC BY-SA 4.0 via Wikimedia Commons.
Bibliography
50
Barnett, Raymond, Michael Ziegler, Karl Byleen, and David Sobecki. College Algebra with Trigonometry. Boston:
McGraw Hill Higher Education, 2008.
Bittinger, Marvin L., Judith A. Beecher, David J. Ellenbogen, and Judith A. Penna. Algebra and Trigonometry:
Graphs and Models. 4th ed. Boston: Pearson/Addison Wesley, 2009.
Blitzer, Robert. Algebra and Trigonometry. 3rd ed. Upper Saddle River, New Jersey: Pearson/Prentice Hal, 2007.
Bourne, Murray. 6. The Hyperbola. Interactive Mathematics. Accessed from https://www.intmath.com/plane-
analytic-geometry/6-hyperbola.php. March 26, 2020.
Larson, Ron. College Algebra with Applications for Business and the Life Sciences. Boston: MA: Houghton Mifflin,
2009.
OpenStax College. Solving Applied Problems Involving Hyperbolas. Lumen Learning. Accessed from
https://courses.lumenlearning.com/ivytech-collegealgebra/chapter/solving-applied-problems-involving-
hyperbolas/. March 26, 2020.
Simmons, George F. Calculus with Analytic Geometry. 2nd ed. New York: McGraw-Hill, 1996.

More Related Content

What's hot

Chapter 4 part3- Means and Variances of Random Variables
Chapter 4 part3- Means and Variances of Random VariablesChapter 4 part3- Means and Variances of Random Variables
Chapter 4 part3- Means and Variances of Random Variables
nszakir
 
10.1 Distance and Midpoint Formulas
10.1 Distance and Midpoint Formulas10.1 Distance and Midpoint Formulas
10.1 Distance and Midpoint Formulas
swartzje
 
If and then statements
If and then statementsIf and then statements
If and then statements
ValDarylAnhao2
 
Conic sections circles - STEM TEACH
Conic sections circles - STEM TEACHConic sections circles - STEM TEACH
Conic sections circles - STEM TEACH
Mr Math
 
The Fundamental Counting Principle
The Fundamental Counting PrincipleThe Fundamental Counting Principle
The Fundamental Counting Principle
Ron Eick
 
Factoring the greatest common monomial factor
Factoring the greatest common monomial factorFactoring the greatest common monomial factor
Factoring the greatest common monomial factor
Nara Cocarelli
 
Lecture 03 special products and factoring
Lecture 03 special products and factoringLecture 03 special products and factoring
Lecture 03 special products and factoring
Hazel Joy Chong
 
Grade 7 Mathematics Week 4 2nd Quarter
Grade 7 Mathematics Week 4 2nd QuarterGrade 7 Mathematics Week 4 2nd Quarter
Grade 7 Mathematics Week 4 2nd Quarter
jennytuazon01630
 
Geometric Sequence
Geometric SequenceGeometric Sequence
Geometric Sequence
Fe Lago
 
11 2 arcs and central angles lesson
11 2 arcs and central angles lesson11 2 arcs and central angles lesson
11 2 arcs and central angles lesson
gwilson8786
 
Factor Completely Different Types of Polynomials
Factor Completely Different Types of PolynomialsFactor Completely Different Types of Polynomials
Factor Completely Different Types of Polynomials
AleisandraBunayog
 
Fundamental counting principle powerpoint
Fundamental counting principle powerpointFundamental counting principle powerpoint
Fundamental counting principle powerpoint
mesmith1
 
Geometric series
Geometric seriesGeometric series
Geometric series
Jhon Paul Lagumbay
 
Exponential Equation & Inequalities.pptx
Exponential Equation & Inequalities.pptxExponential Equation & Inequalities.pptx
Exponential Equation & Inequalities.pptx
Roqui Gonzaga
 
Sample space, events, outcomes, and experiments
Sample space, events, outcomes, and experimentsSample space, events, outcomes, and experiments
Sample space, events, outcomes, and experiments
Christian Costa
 
Chapter 1 random variables and probability distributions
Chapter 1   random variables and probability distributionsChapter 1   random variables and probability distributions
Chapter 1 random variables and probability distributions
Antonio F. Balatar Jr.
 
Parallel And Perpendicular Lines
Parallel And Perpendicular LinesParallel And Perpendicular Lines
Parallel And Perpendicular Lines
guestd1dc2e
 
union and intersection of events.ppt
union and intersection of events.pptunion and intersection of events.ppt
union and intersection of events.ppt
Izah Catli
 
Domain-Range-Intercepts-Zeros-and-Asymptotes-of-Rational-Function.pptx
Domain-Range-Intercepts-Zeros-and-Asymptotes-of-Rational-Function.pptxDomain-Range-Intercepts-Zeros-and-Asymptotes-of-Rational-Function.pptx
Domain-Range-Intercepts-Zeros-and-Asymptotes-of-Rational-Function.pptx
NeomyAngelaLeono1
 
Introduction to Polynomial Functions
Introduction to Polynomial FunctionsIntroduction to Polynomial Functions
Introduction to Polynomial Functions
kshoskey
 

What's hot (20)

Chapter 4 part3- Means and Variances of Random Variables
Chapter 4 part3- Means and Variances of Random VariablesChapter 4 part3- Means and Variances of Random Variables
Chapter 4 part3- Means and Variances of Random Variables
 
10.1 Distance and Midpoint Formulas
10.1 Distance and Midpoint Formulas10.1 Distance and Midpoint Formulas
10.1 Distance and Midpoint Formulas
 
If and then statements
If and then statementsIf and then statements
If and then statements
 
Conic sections circles - STEM TEACH
Conic sections circles - STEM TEACHConic sections circles - STEM TEACH
Conic sections circles - STEM TEACH
 
The Fundamental Counting Principle
The Fundamental Counting PrincipleThe Fundamental Counting Principle
The Fundamental Counting Principle
 
Factoring the greatest common monomial factor
Factoring the greatest common monomial factorFactoring the greatest common monomial factor
Factoring the greatest common monomial factor
 
Lecture 03 special products and factoring
Lecture 03 special products and factoringLecture 03 special products and factoring
Lecture 03 special products and factoring
 
Grade 7 Mathematics Week 4 2nd Quarter
Grade 7 Mathematics Week 4 2nd QuarterGrade 7 Mathematics Week 4 2nd Quarter
Grade 7 Mathematics Week 4 2nd Quarter
 
Geometric Sequence
Geometric SequenceGeometric Sequence
Geometric Sequence
 
11 2 arcs and central angles lesson
11 2 arcs and central angles lesson11 2 arcs and central angles lesson
11 2 arcs and central angles lesson
 
Factor Completely Different Types of Polynomials
Factor Completely Different Types of PolynomialsFactor Completely Different Types of Polynomials
Factor Completely Different Types of Polynomials
 
Fundamental counting principle powerpoint
Fundamental counting principle powerpointFundamental counting principle powerpoint
Fundamental counting principle powerpoint
 
Geometric series
Geometric seriesGeometric series
Geometric series
 
Exponential Equation & Inequalities.pptx
Exponential Equation & Inequalities.pptxExponential Equation & Inequalities.pptx
Exponential Equation & Inequalities.pptx
 
Sample space, events, outcomes, and experiments
Sample space, events, outcomes, and experimentsSample space, events, outcomes, and experiments
Sample space, events, outcomes, and experiments
 
Chapter 1 random variables and probability distributions
Chapter 1   random variables and probability distributionsChapter 1   random variables and probability distributions
Chapter 1 random variables and probability distributions
 
Parallel And Perpendicular Lines
Parallel And Perpendicular LinesParallel And Perpendicular Lines
Parallel And Perpendicular Lines
 
union and intersection of events.ppt
union and intersection of events.pptunion and intersection of events.ppt
union and intersection of events.ppt
 
Domain-Range-Intercepts-Zeros-and-Asymptotes-of-Rational-Function.pptx
Domain-Range-Intercepts-Zeros-and-Asymptotes-of-Rational-Function.pptxDomain-Range-Intercepts-Zeros-and-Asymptotes-of-Rational-Function.pptx
Domain-Range-Intercepts-Zeros-and-Asymptotes-of-Rational-Function.pptx
 
Introduction to Polynomial Functions
Introduction to Polynomial FunctionsIntroduction to Polynomial Functions
Introduction to Polynomial Functions
 

Similar to PRE- CALCULUS

PLANE-TRIGONOMETRY.pptx
PLANE-TRIGONOMETRY.pptxPLANE-TRIGONOMETRY.pptx
PLANE-TRIGONOMETRY.pptx
DanEmilBernardino
 
Arc Length & Area of a Sector.pptx
Arc Length & Area of a Sector.pptxArc Length & Area of a Sector.pptx
Arc Length & Area of a Sector.pptx
JASONBIGATA4
 
Antenna course sofia_2015_lisi_lesson1_v03
Antenna course sofia_2015_lisi_lesson1_v03Antenna course sofia_2015_lisi_lesson1_v03
Antenna course sofia_2015_lisi_lesson1_v03
Marco Lisi
 
Cirlces day 1 and day 2 together
Cirlces day 1 and day 2 togetherCirlces day 1 and day 2 together
Cirlces day 1 and day 2 together
jbianco9910
 
Arc Length & Area of a Sector.pptx
Arc Length & Area of a Sector.pptxArc Length & Area of a Sector.pptx
Arc Length & Area of a Sector.pptx
JASONBIGATA4
 
Conic sections in daily life
Conic sections in daily lifeConic sections in daily life
Conic sections in daily life
Aditya Garg
 
Ir spectroscopy from nstu
Ir spectroscopy from nstuIr spectroscopy from nstu
Ir spectroscopy from nstu
Arafat Jakir
 
GENCHEM1-LESSON 1-2-QUANTUM NUMBERS.pptx
GENCHEM1-LESSON 1-2-QUANTUM NUMBERS.pptxGENCHEM1-LESSON 1-2-QUANTUM NUMBERS.pptx
GENCHEM1-LESSON 1-2-QUANTUM NUMBERS.pptx
SirMiguelMalvar
 
Fundamentals of Trigonometry.pdf
Fundamentals of Trigonometry.pdfFundamentals of Trigonometry.pdf
Fundamentals of Trigonometry.pdf
NusratIqbal9
 
Apperture and Horn Antenna
Apperture and Horn AntennaApperture and Horn Antenna
Apperture and Horn Antenna
Roma Rico Flores
 
angle_of_elevation_depression cot2.pptx
angle_of_elevation_depression cot2.pptxangle_of_elevation_depression cot2.pptx
angle_of_elevation_depression cot2.pptx
Vallerie Gonzales-lebaste
 
Scientific notation
Scientific notationScientific notation
Scientific notation
Yann Villarreal
 
1-ELLIPSE.pptx
1-ELLIPSE.pptx1-ELLIPSE.pptx
1-ELLIPSE.pptx
JohannPaulusRobles
 
Geometry unit 8.4
Geometry unit 8.4Geometry unit 8.4
Geometry unit 8.4
Mark Ryder
 
Indoor Heading Estimation
 Indoor Heading Estimation Indoor Heading Estimation
Indoor Heading Estimation
Alwin Poulose
 
(1) angular m easure
(1) angular m easure(1) angular m easure
(1) angular m easure
physics101
 
Conic Sections (Class 11 Project)
Conic Sections (Class 11 Project)Conic Sections (Class 11 Project)
Conic Sections (Class 11 Project)
KushagraAgrawal46
 
Situational-Problem-Involving-Conic-Sections.pptx
Situational-Problem-Involving-Conic-Sections.pptxSituational-Problem-Involving-Conic-Sections.pptx
Situational-Problem-Involving-Conic-Sections.pptx
AmeeraHananHamza
 
Pre c alc module 1-conic-sections
Pre c alc module 1-conic-sectionsPre c alc module 1-conic-sections
Pre c alc module 1-conic-sections
Eclaro College
 
1-PHYSICAL QUANTITIES, UNITS & MEASUREMENT131029195248-phpapp01.ppt
1-PHYSICAL QUANTITIES, UNITS & MEASUREMENT131029195248-phpapp01.ppt1-PHYSICAL QUANTITIES, UNITS & MEASUREMENT131029195248-phpapp01.ppt
1-PHYSICAL QUANTITIES, UNITS & MEASUREMENT131029195248-phpapp01.ppt
ChenKahPin
 

Similar to PRE- CALCULUS (20)

PLANE-TRIGONOMETRY.pptx
PLANE-TRIGONOMETRY.pptxPLANE-TRIGONOMETRY.pptx
PLANE-TRIGONOMETRY.pptx
 
Arc Length & Area of a Sector.pptx
Arc Length & Area of a Sector.pptxArc Length & Area of a Sector.pptx
Arc Length & Area of a Sector.pptx
 
Antenna course sofia_2015_lisi_lesson1_v03
Antenna course sofia_2015_lisi_lesson1_v03Antenna course sofia_2015_lisi_lesson1_v03
Antenna course sofia_2015_lisi_lesson1_v03
 
Cirlces day 1 and day 2 together
Cirlces day 1 and day 2 togetherCirlces day 1 and day 2 together
Cirlces day 1 and day 2 together
 
Arc Length & Area of a Sector.pptx
Arc Length & Area of a Sector.pptxArc Length & Area of a Sector.pptx
Arc Length & Area of a Sector.pptx
 
Conic sections in daily life
Conic sections in daily lifeConic sections in daily life
Conic sections in daily life
 
Ir spectroscopy from nstu
Ir spectroscopy from nstuIr spectroscopy from nstu
Ir spectroscopy from nstu
 
GENCHEM1-LESSON 1-2-QUANTUM NUMBERS.pptx
GENCHEM1-LESSON 1-2-QUANTUM NUMBERS.pptxGENCHEM1-LESSON 1-2-QUANTUM NUMBERS.pptx
GENCHEM1-LESSON 1-2-QUANTUM NUMBERS.pptx
 
Fundamentals of Trigonometry.pdf
Fundamentals of Trigonometry.pdfFundamentals of Trigonometry.pdf
Fundamentals of Trigonometry.pdf
 
Apperture and Horn Antenna
Apperture and Horn AntennaApperture and Horn Antenna
Apperture and Horn Antenna
 
angle_of_elevation_depression cot2.pptx
angle_of_elevation_depression cot2.pptxangle_of_elevation_depression cot2.pptx
angle_of_elevation_depression cot2.pptx
 
Scientific notation
Scientific notationScientific notation
Scientific notation
 
1-ELLIPSE.pptx
1-ELLIPSE.pptx1-ELLIPSE.pptx
1-ELLIPSE.pptx
 
Geometry unit 8.4
Geometry unit 8.4Geometry unit 8.4
Geometry unit 8.4
 
Indoor Heading Estimation
 Indoor Heading Estimation Indoor Heading Estimation
Indoor Heading Estimation
 
(1) angular m easure
(1) angular m easure(1) angular m easure
(1) angular m easure
 
Conic Sections (Class 11 Project)
Conic Sections (Class 11 Project)Conic Sections (Class 11 Project)
Conic Sections (Class 11 Project)
 
Situational-Problem-Involving-Conic-Sections.pptx
Situational-Problem-Involving-Conic-Sections.pptxSituational-Problem-Involving-Conic-Sections.pptx
Situational-Problem-Involving-Conic-Sections.pptx
 
Pre c alc module 1-conic-sections
Pre c alc module 1-conic-sectionsPre c alc module 1-conic-sections
Pre c alc module 1-conic-sections
 
1-PHYSICAL QUANTITIES, UNITS & MEASUREMENT131029195248-phpapp01.ppt
1-PHYSICAL QUANTITIES, UNITS & MEASUREMENT131029195248-phpapp01.ppt1-PHYSICAL QUANTITIES, UNITS & MEASUREMENT131029195248-phpapp01.ppt
1-PHYSICAL QUANTITIES, UNITS & MEASUREMENT131029195248-phpapp01.ppt
 

More from Jasten Domingo

Monthly Holy Mass - Pagdiriwang ng Banal na Eukaristiya
Monthly Holy Mass - Pagdiriwang ng Banal na EukaristiyaMonthly Holy Mass - Pagdiriwang ng Banal na Eukaristiya
Monthly Holy Mass - Pagdiriwang ng Banal na Eukaristiya
Jasten Domingo
 
Confucianism
ConfucianismConfucianism
Confucianism
Jasten Domingo
 
Mahayana buddhism
Mahayana buddhismMahayana buddhism
Mahayana buddhism
Jasten Domingo
 
Theravada Buddhism
Theravada BuddhismTheravada Buddhism
Theravada Buddhism
Jasten Domingo
 
Islam
IslamIslam
Christianity
ChristianityChristianity
Christianity
Jasten Domingo
 

More from Jasten Domingo (6)

Monthly Holy Mass - Pagdiriwang ng Banal na Eukaristiya
Monthly Holy Mass - Pagdiriwang ng Banal na EukaristiyaMonthly Holy Mass - Pagdiriwang ng Banal na Eukaristiya
Monthly Holy Mass - Pagdiriwang ng Banal na Eukaristiya
 
Confucianism
ConfucianismConfucianism
Confucianism
 
Mahayana buddhism
Mahayana buddhismMahayana buddhism
Mahayana buddhism
 
Theravada Buddhism
Theravada BuddhismTheravada Buddhism
Theravada Buddhism
 
Islam
IslamIslam
Islam
 
Christianity
ChristianityChristianity
Christianity
 

Recently uploaded

Haunted Houses by H W Longfellow for class 10
Haunted Houses by H W Longfellow for class 10Haunted Houses by H W Longfellow for class 10
Haunted Houses by H W Longfellow for class 10
nitinpv4ai
 
Creative Restart 2024: Mike Martin - Finding a way around “no”
Creative Restart 2024: Mike Martin - Finding a way around “no”Creative Restart 2024: Mike Martin - Finding a way around “no”
Creative Restart 2024: Mike Martin - Finding a way around “no”
Taste
 
Accounting for Restricted Grants When and How To Record Properly
Accounting for Restricted Grants  When and How To Record ProperlyAccounting for Restricted Grants  When and How To Record Properly
Accounting for Restricted Grants When and How To Record Properly
TechSoup
 
INTRODUCTION TO HOSPITALS & AND ITS ORGANIZATION
INTRODUCTION TO HOSPITALS & AND ITS ORGANIZATION INTRODUCTION TO HOSPITALS & AND ITS ORGANIZATION
INTRODUCTION TO HOSPITALS & AND ITS ORGANIZATION
ShwetaGawande8
 
How to Download & Install Module From the Odoo App Store in Odoo 17
How to Download & Install Module From the Odoo App Store in Odoo 17How to Download & Install Module From the Odoo App Store in Odoo 17
How to Download & Install Module From the Odoo App Store in Odoo 17
Celine George
 
Ch-4 Forest Society and colonialism 2.pdf
Ch-4 Forest Society and colonialism 2.pdfCh-4 Forest Society and colonialism 2.pdf
Ch-4 Forest Society and colonialism 2.pdf
lakshayrojroj
 
Creation or Update of a Mandatory Field is Not Set in Odoo 17
Creation or Update of a Mandatory Field is Not Set in Odoo 17Creation or Update of a Mandatory Field is Not Set in Odoo 17
Creation or Update of a Mandatory Field is Not Set in Odoo 17
Celine George
 
The basics of sentences session 7pptx.pptx
The basics of sentences session 7pptx.pptxThe basics of sentences session 7pptx.pptx
The basics of sentences session 7pptx.pptx
heathfieldcps1
 
HYPERTENSION - SLIDE SHARE PRESENTATION.
HYPERTENSION - SLIDE SHARE PRESENTATION.HYPERTENSION - SLIDE SHARE PRESENTATION.
HYPERTENSION - SLIDE SHARE PRESENTATION.
deepaannamalai16
 
adjectives.ppt for class 1 to 6, grammar
adjectives.ppt for class 1 to 6, grammaradjectives.ppt for class 1 to 6, grammar
adjectives.ppt for class 1 to 6, grammar
7DFarhanaMohammed
 
How to Manage Reception Report in Odoo 17
How to Manage Reception Report in Odoo 17How to Manage Reception Report in Odoo 17
How to Manage Reception Report in Odoo 17
Celine George
 
Educational Technology in the Health Sciences
Educational Technology in the Health SciencesEducational Technology in the Health Sciences
Educational Technology in the Health Sciences
Iris Thiele Isip-Tan
 
220711130097 Tulip Samanta Concept of Information and Communication Technology
220711130097 Tulip Samanta Concept of Information and Communication Technology220711130097 Tulip Samanta Concept of Information and Communication Technology
220711130097 Tulip Samanta Concept of Information and Communication Technology
Kalna College
 
欧洲杯下注-欧洲杯下注押注官网-欧洲杯下注押注网站|【​网址​🎉ac44.net🎉​】
欧洲杯下注-欧洲杯下注押注官网-欧洲杯下注押注网站|【​网址​🎉ac44.net🎉​】欧洲杯下注-欧洲杯下注押注官网-欧洲杯下注押注网站|【​网址​🎉ac44.net🎉​】
欧洲杯下注-欧洲杯下注押注官网-欧洲杯下注押注网站|【​网址​🎉ac44.net🎉​】
andagarcia212
 
Standardized tool for Intelligence test.
Standardized tool for Intelligence test.Standardized tool for Intelligence test.
Standardized tool for Intelligence test.
deepaannamalai16
 
78 Microsoft-Publisher - Sirin Sultana Bora.pptx
78 Microsoft-Publisher - Sirin Sultana Bora.pptx78 Microsoft-Publisher - Sirin Sultana Bora.pptx
78 Microsoft-Publisher - Sirin Sultana Bora.pptx
Kalna College
 
Diversity Quiz Prelims by Quiz Club, IIT Kanpur
Diversity Quiz Prelims by Quiz Club, IIT KanpurDiversity Quiz Prelims by Quiz Club, IIT Kanpur
Diversity Quiz Prelims by Quiz Club, IIT Kanpur
Quiz Club IIT Kanpur
 
Information and Communication Technology in Education
Information and Communication Technology in EducationInformation and Communication Technology in Education
Information and Communication Technology in Education
MJDuyan
 
Andreas Schleicher presents PISA 2022 Volume III - Creative Thinking - 18 Jun...
Andreas Schleicher presents PISA 2022 Volume III - Creative Thinking - 18 Jun...Andreas Schleicher presents PISA 2022 Volume III - Creative Thinking - 18 Jun...
Andreas Schleicher presents PISA 2022 Volume III - Creative Thinking - 18 Jun...
EduSkills OECD
 
220711130088 Sumi Basak Virtual University EPC 3.pptx
220711130088 Sumi Basak Virtual University EPC 3.pptx220711130088 Sumi Basak Virtual University EPC 3.pptx
220711130088 Sumi Basak Virtual University EPC 3.pptx
Kalna College
 

Recently uploaded (20)

Haunted Houses by H W Longfellow for class 10
Haunted Houses by H W Longfellow for class 10Haunted Houses by H W Longfellow for class 10
Haunted Houses by H W Longfellow for class 10
 
Creative Restart 2024: Mike Martin - Finding a way around “no”
Creative Restart 2024: Mike Martin - Finding a way around “no”Creative Restart 2024: Mike Martin - Finding a way around “no”
Creative Restart 2024: Mike Martin - Finding a way around “no”
 
Accounting for Restricted Grants When and How To Record Properly
Accounting for Restricted Grants  When and How To Record ProperlyAccounting for Restricted Grants  When and How To Record Properly
Accounting for Restricted Grants When and How To Record Properly
 
INTRODUCTION TO HOSPITALS & AND ITS ORGANIZATION
INTRODUCTION TO HOSPITALS & AND ITS ORGANIZATION INTRODUCTION TO HOSPITALS & AND ITS ORGANIZATION
INTRODUCTION TO HOSPITALS & AND ITS ORGANIZATION
 
How to Download & Install Module From the Odoo App Store in Odoo 17
How to Download & Install Module From the Odoo App Store in Odoo 17How to Download & Install Module From the Odoo App Store in Odoo 17
How to Download & Install Module From the Odoo App Store in Odoo 17
 
Ch-4 Forest Society and colonialism 2.pdf
Ch-4 Forest Society and colonialism 2.pdfCh-4 Forest Society and colonialism 2.pdf
Ch-4 Forest Society and colonialism 2.pdf
 
Creation or Update of a Mandatory Field is Not Set in Odoo 17
Creation or Update of a Mandatory Field is Not Set in Odoo 17Creation or Update of a Mandatory Field is Not Set in Odoo 17
Creation or Update of a Mandatory Field is Not Set in Odoo 17
 
The basics of sentences session 7pptx.pptx
The basics of sentences session 7pptx.pptxThe basics of sentences session 7pptx.pptx
The basics of sentences session 7pptx.pptx
 
HYPERTENSION - SLIDE SHARE PRESENTATION.
HYPERTENSION - SLIDE SHARE PRESENTATION.HYPERTENSION - SLIDE SHARE PRESENTATION.
HYPERTENSION - SLIDE SHARE PRESENTATION.
 
adjectives.ppt for class 1 to 6, grammar
adjectives.ppt for class 1 to 6, grammaradjectives.ppt for class 1 to 6, grammar
adjectives.ppt for class 1 to 6, grammar
 
How to Manage Reception Report in Odoo 17
How to Manage Reception Report in Odoo 17How to Manage Reception Report in Odoo 17
How to Manage Reception Report in Odoo 17
 
Educational Technology in the Health Sciences
Educational Technology in the Health SciencesEducational Technology in the Health Sciences
Educational Technology in the Health Sciences
 
220711130097 Tulip Samanta Concept of Information and Communication Technology
220711130097 Tulip Samanta Concept of Information and Communication Technology220711130097 Tulip Samanta Concept of Information and Communication Technology
220711130097 Tulip Samanta Concept of Information and Communication Technology
 
欧洲杯下注-欧洲杯下注押注官网-欧洲杯下注押注网站|【​网址​🎉ac44.net🎉​】
欧洲杯下注-欧洲杯下注押注官网-欧洲杯下注押注网站|【​网址​🎉ac44.net🎉​】欧洲杯下注-欧洲杯下注押注官网-欧洲杯下注押注网站|【​网址​🎉ac44.net🎉​】
欧洲杯下注-欧洲杯下注押注官网-欧洲杯下注押注网站|【​网址​🎉ac44.net🎉​】
 
Standardized tool for Intelligence test.
Standardized tool for Intelligence test.Standardized tool for Intelligence test.
Standardized tool for Intelligence test.
 
78 Microsoft-Publisher - Sirin Sultana Bora.pptx
78 Microsoft-Publisher - Sirin Sultana Bora.pptx78 Microsoft-Publisher - Sirin Sultana Bora.pptx
78 Microsoft-Publisher - Sirin Sultana Bora.pptx
 
Diversity Quiz Prelims by Quiz Club, IIT Kanpur
Diversity Quiz Prelims by Quiz Club, IIT KanpurDiversity Quiz Prelims by Quiz Club, IIT Kanpur
Diversity Quiz Prelims by Quiz Club, IIT Kanpur
 
Information and Communication Technology in Education
Information and Communication Technology in EducationInformation and Communication Technology in Education
Information and Communication Technology in Education
 
Andreas Schleicher presents PISA 2022 Volume III - Creative Thinking - 18 Jun...
Andreas Schleicher presents PISA 2022 Volume III - Creative Thinking - 18 Jun...Andreas Schleicher presents PISA 2022 Volume III - Creative Thinking - 18 Jun...
Andreas Schleicher presents PISA 2022 Volume III - Creative Thinking - 18 Jun...
 
220711130088 Sumi Basak Virtual University EPC 3.pptx
220711130088 Sumi Basak Virtual University EPC 3.pptx220711130088 Sumi Basak Virtual University EPC 3.pptx
220711130088 Sumi Basak Virtual University EPC 3.pptx
 

PRE- CALCULUS

  • 1. Capstone Project Science, Technology, Engineering, and Mathematics Precalculus Science, Technology, Engineering, and Mathematics Lesson 4.3 Applications of Hyperbolas in Real-life Situations
  • 2. 2 Have you every wondered how we were able to track the locations of ships and aircrafts before we had navigation satellites (e.g. Global Positioning System or GPS)?
  • 3. 3 Navigation systems such as the LORAN (long-range navigation) use radio signals to determine the exact location of a ship or aircraft.
  • 4. 4 Knowledge of hyperbolas is used in this type of navigation system.
  • 5. 5 What are some real-life applications of hyperbolas?
  • 6. Learning Competency At the end of the lesson, you should be able to do the following: 6 Solve situational problems involving hyperbolas (STEM_PC11AG-Ie-2).
  • 7. Learning Objectives At the end of the lesson, you should be able to do the following: 7 ● Recall the different parts and properties of a hyperbola. ● Solve word problems on hyperbolas.
  • 8. 8 A hyperbola is defined as a set of points on a plane whose absolute difference between the distances from the foci, 𝐹1 and 𝐹2, is constant. Hyperbolas
  • 9. 9 Parts and Properties of Hyperbolas Given arbitrary points 𝑃1 and 𝑃2 on the hyperbola, |𝑃1𝐹1 − 𝑃1𝐹2| = 𝑃2𝐹1 − 𝑃2𝐹2 = 2𝑎. What is the distance 𝑎?
  • 10. 10 Parts and Properties of Hyperbolas 𝒄2 = 𝒂2 + 𝒃2
  • 11. 11 How do you determine the value of 𝒂 given the values of 𝒃 and 𝒄? How do you determine the value of 𝒃 given the values of 𝒂 and 𝒄?
  • 12. 12 Standard Equations of a Hyperbola Orientation of Principal Axis Equation Horizontal 𝒙 − 𝒉 𝟐 𝒂𝟐 − 𝒚 − 𝒌 𝟐 𝒃𝟐 = 𝟏 Vertical 𝒚 − 𝒌 𝟐 𝒂𝟐 − 𝒙 − 𝒉 𝟐 𝒃𝟐 = 𝟏
  • 14. 14 Applications of Hyperbolas Hyperbolic Navigation Two stations, 𝐴 and 𝐵, transmit radio signals. The radio signals reach the ship at different times.
  • 15. 15 Applications of Hyperbolas Hyperbolic Navigation The location 𝑃 of the ship is somewhere on a hyperbola.
  • 16. 16 Applications of Hyperbolas Hyperbolic Navigation The difference between the distances of the two stations from the ship at point 𝑃 is 𝑷𝑨 − 𝑷𝑩 = 𝟐𝒂.
  • 17. 17 Applications of Hyperbolas Cooling Towers Cooling towers use water from rivers and lakes to cool nuclear power plants.
  • 18. 18 Applications of Hyperbolas Cooling Towers They protect aquatic life by ensuring that the water is returned to the environment at normal temperatures.
  • 19. 19 Applications of Hyperbolas Cooling Towers Cooling towers are hyperboloid in shape. Why?
  • 20. 20 Applications of Hyperbolas Cooling Towers  able to withstand strong winds  efficient in cooling  economical
  • 21. Let’s Practice! 21 Point 𝑷 is on a hyperbola. The distance of 𝑷 from the first focus is 6 units more than its distance from the second focus. What is the distance of the center to a vertex of the hyperbola?
  • 22. Let’s Practice! 22 3 units Point 𝑷 is on a hyperbola. The distance of 𝑷 from the first focus is 6 units more than its distance from the second focus. What is the distance of the center to a vertex of the hyperbola?
  • 23. Try It! 23 23 Point 𝑷 is on a hyperbola. The difference between the distances of 𝑷 from the first focus and the second focus is 20 units. What is the distance of the center to a vertex of the hyperbola?
  • 24. Let’s Practice! 24 What is the equation of a hyperbola whose center is at the origin, has a horizontal principal axis, the value of 𝒂 is 𝟒, and the point (𝟗, 𝟏𝟒) is on the hyperbola?
  • 25. Let’s Practice! 25 𝒙𝟐 𝟏𝟔 − 𝒚𝟐 𝟒𝟖. 𝟐𝟓 = 𝟏 What is the equation of a hyperbola whose center is at the origin, has a horizontal principal axis, the value of 𝒂 is 𝟒, and the point (𝟗, 𝟏𝟒) is on the hyperbola?
  • 26. Try It! 26 26 What is the equation of a hyperbola whose center is at the origin, has a vertical principal axis, the value of 𝒂 is 𝟖, and the point (𝟖, 𝟏𝟑) is on the hyperbola?
  • 27. Let’s Practice! 27 Point 𝑷 is on a hyperbola with a vertical principal axis and whose center is at the origin. The distance of 𝑷 from the first focus 𝑭𝟏 is 12 units more than its distance from the second focus 𝑭𝟐. The distance between the two foci is 20 units. What is the equation of the hyperbola?
  • 28. Let’s Practice! 28 𝒚𝟐 𝟑𝟔 − 𝒙𝟐 𝟔𝟒 = 𝟏 Point 𝑷 is on a hyperbola with a vertical principal axis and whose center is at the origin. The distance of 𝑷 from the first focus 𝑭𝟏 is 12 units more than its distance from the second focus 𝑭𝟐. The distance between the two foci is 20 units. What is the equation of the hyperbola?
  • 29. Try It! 29 29 Point 𝑷 is on a hyperbola with a vertical principal axis and whose center is at the origin. The distance of 𝑷 from the first focus 𝑭𝟏 is 10 units more than its distance from the second focus 𝑭𝟐. The distance between the two foci is 26 units. What is the equation of the hyperbola?
  • 30. Let’s Practice! 30 The sides of a cooling tower represent a hyperbola. The width of the slimmest part of the cooling tower is 50 meters. The length from this point to the top is 64 meters. The diameter of the top of the cooling tower is 60 meters. Find the equation of the hyperbola that represents the sides of the cooling tower. Set the middle of the slimmest part as the origin.
  • 31. Let’s Practice! 31 𝒙𝟐 𝟔𝟐𝟓 − 𝒚𝟐 𝟗𝟑𝟎𝟗. 𝟎𝟗 = 𝟏 The sides of a cooling tower represent a hyperbola. The width of the slimmest part of the cooling tower is 50 meters. The length from this point to the top is 64 meters. The diameter of the top of the cooling tower is 60 meters. Find the equation of the hyperbola that represents the sides of the cooling tower. Set the middle of the slimmest part as the origin.
  • 32. Try It! 32 32 The sides of a cooling tower represent a hyperbola. The width of the slimmest part of the cooling tower is 54 meters. The length from this point to the top is 70 meters. The diameter of the top of the cooling tower is 68 meters. Find the equation of the hyperbola that represents the sides of the cooling tower. Set the middle of the slimmest part as the origin.
  • 33. Let’s Practice! 33 Two stations 𝑨 and 𝑩 are along a straight coast such that station 𝑨 is 100 miles west of station 𝑩. They both transmit radio signals at the speed of 186 000 miles per second. A ship sailing at sea is 50 miles from the coast. The radio signal from station 𝑩 arrives at the ship 0.0002 of a second earlier than the radio signal sent from station 𝑨. Where is the ship? Set the midpoint of 𝑨 and 𝑩 as the origin.
  • 34. Let’s Practice! 34 (𝟐𝟕. 𝟑𝟒, 𝟓𝟎) Two stations 𝑨 and 𝑩 are along a straight coast such that station 𝑨 is 100 miles west of station 𝑩. They both transmit radio signals at the speed of 186 000 miles per second. A ship sailing at sea is 50 miles from the coast. The radio signal from station 𝑩 arrives at the ship 0.0002 of a second earlier than the radio signal sent from station 𝑨. Where is the ship? Set the midpoint of 𝑨 and 𝑩 as the origin.
  • 35. Try It! 35 35 Two stations 𝑨 and 𝑩 transmit radio signals such that station A is 200 miles west of station 𝑩. Both stations sent radio signals with a speed of 𝟎. 𝟐 miles per microsecond to a plane traveling west. The signal from station 𝑩 reaches a plane 500 microseconds faster than the signal from station 𝑨. If the plane is 80 miles north of the line from stations 𝑨 to 𝑩, what is the location of the plane? Set the midpoint of 𝑨 and 𝑩 as the origin.
  • 36. Check Your Understanding 36 Let 𝑷 be a point on a hyperbola with center at the origin and foci 𝑭𝟏 and 𝑭𝟐. Answer the following questions. Round off your answer to two decimal places. 1. If 𝑃𝐹1 − 𝑃𝐹2 = 18, what is 𝑎? 2. If 𝑃𝐹2 − 𝑃𝐹1 = 32 and 𝑏 = 12, what is 𝑐? 3. If 𝑃𝐹1 − 𝑃𝐹2 = 34 and 𝑏2 = 256, what is 𝑐? 4. If 𝑐 = 39 and 𝑎 = 36, what is 𝑏? 5. If 𝑐2 = 1 600 and 𝑎 = 32, what is 𝑏?
  • 37. Check Your Understanding 37 Solve the following problems. 1. A nuclear power plant has a cooling tower whose sides are in the shape of a hyperbola. The slimmest part of the tower is 58 meters. The length from this point to the top is 80 meters. The radius of the top of the cooling tower is 72 meters. What is the equation of the hyperbola that represents the sides of the cooling tower? Set the middle of the slimmest part as the origin.
  • 38. Check Your Understanding 38 Solve the following problems. 2. The sides of a cooling tower represent a hyperbola. The width of the slimmest part of the cooling tower is 80 meters. The slimmest part of the tower is 90 meters from the ground. The diameter of the base of the tower is 130 meters. What is the equation of the hyperbola that represents the sides of the cooling tower? Set the middle of the slimmest part as the origin.
  • 39. Check Your Understanding 39 Solve the following problems. 3. Two radio signaling stations 𝐴 and 𝐵 are 120 kilometers apart. The radio signal from the two stations has a speed of 300 000 kilometers per second. A ship at sea receives the signals such that the signal from station 𝐵 arrives 0.0002 seconds before the signal from station 𝐴. What is the equation of the hyperbola where the ship is located? Set the midpoint of 𝐴 and 𝐵 as the origin.
  • 40. Let’s Sum It Up! 40 ● A hyperbola is a set of points on a plane whose absolute difference between the distances from two fixed points 𝐹1 and 𝐹2 is constant. ● Given any point 𝑃 on a hyperbola with foci 𝐹1 and 𝐹2, 𝑃𝐹1 − 𝑃𝐹2 = 2𝑎.
  • 41. Let’s Sum It Up! 41 ● The distance from the center to a vertex of the hyperbola is 𝒂, the distance from the center to an endpoint of the conjugate axis is 𝒃, and the focal distance is 𝒄. ● The relationship between the three distances is 𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐.
  • 42. Let’s Sum It Up! 42 ● The standard forms of equation of a hyperbola whose center is at the origin are 𝒙𝟐 𝒂𝟐 − 𝒚𝟐 𝒃𝟐 = 𝟏 if the principal axis is horizontal, and 𝒚𝟐 𝒂𝟐 − 𝒙𝟐 𝒃𝟐 = 𝟏 if the principal axis is vertical.
  • 43. Let’s Sum It Up! 43 ● In the real world, hyperbolas are used for navigation and the structure of cooling towers.
  • 44. Let’s Sum It Up! 44 ● To solve word problems on hyperbolas, follow these steps: 1. Determine the standard form of equation of the hyperbola. 2. Draw an illustration to be able to visualize the problem. 3. Solve for the unknown equation or values.
  • 45. Key Formulas 45 Concept Formula Description Relationship between the values 𝒂, 𝒃, and 𝒄 𝑎2 + 𝑏2 = 𝑐2, where 𝑎 is the distance from the center to a vertex, 𝑏 is the distance from the center to an endpoint of the conjugate axis, and 𝑐 is the focal distance. Use this formula to find an unknown distance (e.g., focal distance) when the other values are known.
  • 46. Key Formulas 46 Concept Formula Description Equation of a Hyperbola in Standard Form 𝑥 − ℎ 2 𝑎2 − 𝑦 − 𝑘 2 𝑏2 = 1, where  (ℎ, 𝑘) is the center;  𝑎 is the distance from the center to a vertex, and  𝑏 is the distance from the center to an endpoint of the conjugate axis. Use this formula to find the equation of a hyperbola given its center, 𝑎, and 𝑏 if the transverse axis is horizontal.
  • 47. Key Formulas 47 Concept Formula Description Equation of a Hyperbola in Standard Form 𝑦 − 𝑘 2 𝑎2 − 𝑥 − ℎ 2 𝑏2 = 1, where  (ℎ, 𝑘) is the center;  𝑎 is the distance from the center to a vertex, and  𝑏 is the distance from the center to an endpoint of the conjugate axis. Use this formula to find the equation of a hyperbola given its center, 𝑎, and 𝑏 if the transverse axis is vertical.
  • 48. Challenge Yourself 48 48 A nuclear power plant has a cooling tower that is hyperboloid in shape. The width of the slimmest part of the cooling tower is 60 meters and is 85 meters from the ground. The diameter of the base of the tower is 120 meters, while the diameter of the top is 96 meters. What is the height of the tower?
  • 49. Photo Credits 49 ● Slides 2 and 3: Navigation boat Engineer Matusevich by Torin is licensed under CC BY-SA 3.0 via Wikimedia Commons. ● Slide 13: Loran C Navigator by Morn the Gorn is licensed under CC BY-SA 3.0 via Wikimedia Commons. ● Slides 17 to 20: Cooling towers of Dukovany Nuclear Power Plant in Dukovany, Třebíč District by Jiří Sedláček is licensed under CC BY-SA 4.0 via Wikimedia Commons.
  • 50. Bibliography 50 Barnett, Raymond, Michael Ziegler, Karl Byleen, and David Sobecki. College Algebra with Trigonometry. Boston: McGraw Hill Higher Education, 2008. Bittinger, Marvin L., Judith A. Beecher, David J. Ellenbogen, and Judith A. Penna. Algebra and Trigonometry: Graphs and Models. 4th ed. Boston: Pearson/Addison Wesley, 2009. Blitzer, Robert. Algebra and Trigonometry. 3rd ed. Upper Saddle River, New Jersey: Pearson/Prentice Hal, 2007. Bourne, Murray. 6. The Hyperbola. Interactive Mathematics. Accessed from https://www.intmath.com/plane- analytic-geometry/6-hyperbola.php. March 26, 2020. Larson, Ron. College Algebra with Applications for Business and the Life Sciences. Boston: MA: Houghton Mifflin, 2009. OpenStax College. Solving Applied Problems Involving Hyperbolas. Lumen Learning. Accessed from https://courses.lumenlearning.com/ivytech-collegealgebra/chapter/solving-applied-problems-involving- hyperbolas/. March 26, 2020. Simmons, George F. Calculus with Analytic Geometry. 2nd ed. New York: McGraw-Hill, 1996.