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A design for a cooling tower project is shown in [link] . Find the equation of the hyperbola that models the sides of the cooling tower. Assume that the center of the hyperbola—indicated by the intersection of dashed perpendicular lines in the figure—is the origin of the coordinate plane. Round final values to four decimal places.

The sides of the tower can be modeled by the hyperbolic equation. x 2 400 y 2 3600 = 1 or  x 2 20 2 y 2 60 2 = 1.

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Key equations

Hyperbola, center at origin, transverse axis on x -axis x 2 a 2 y 2 b 2 = 1
Hyperbola, center at origin, transverse axis on y -axis y 2 a 2 x 2 b 2 = 1
Hyperbola, center at ( h , k ) , transverse axis parallel to x -axis ( x h ) 2 a 2 ( y k ) 2 b 2 = 1
Hyperbola, center at ( h , k ) , transverse axis parallel to y -axis ( y k ) 2 a 2 ( x h ) 2 b 2 = 1

Key concepts

  • A hyperbola is the set of all points ( x , y ) in a plane such that the difference of the distances between ( x , y ) and the foci is a positive constant.
  • The standard form of a hyperbola can be used to locate its vertices and foci. See [link] .
  • When given the coordinates of the foci and vertices of a hyperbola, we can write the equation of the hyperbola in standard form. See [link] and [link] .
  • When given an equation for a hyperbola, we can identify its vertices, co-vertices, foci, asymptotes, and lengths and positions of the transverse and conjugate axes in order to graph the hyperbola. See [link] and [link] .
  • Real-world situations can be modeled using the standard equations of hyperbolas. For instance, given the dimensions of a natural draft cooling tower, we can find a hyperbolic equation that models its sides. See [link] .

Section exercises

Verbal

Define a hyperbola in terms of its foci.

A hyperbola is the set of points in a plane the difference of whose distances from two fixed points (foci) is a positive constant.

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What can we conclude about a hyperbola if its asymptotes intersect at the origin?

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What must be true of the foci of a hyperbola?

The foci must lie on the transverse axis and be in the interior of the hyperbola.

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If the transverse axis of a hyperbola is vertical, what do we know about the graph?

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Where must the center of hyperbola be relative to its foci?

The center must be the midpoint of the line segment joining the foci.

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Algebraic

For the following exercises, determine whether the following equations represent hyperbolas. If so, write in standard form.

x 2 36 y 2 9 = 1

yes x 2 6 2 y 2 3 2 = 1

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25 x 2 16 y 2 = 400

yes x 2 4 2 y 2 5 2 = 1

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9 x 2 + 18 x + y 2 + 4 y 14 = 0

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For the following exercises, write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes.

x 2 25 y 2 36 = 1

x 2 5 2 y 2 6 2 = 1 ; vertices: ( 5 , 0 ) , ( 5 , 0 ) ; foci: ( 61 , 0 ) , ( 61 , 0 ) ; asymptotes: y = 6 5 x , y = 6 5 x

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y 2 4 x 2 81 = 1

y 2 2 2 x 2 9 2 = 1 ; vertices: ( 0 , 2 ) , ( 0 , 2 ) ; foci: ( 0 , 85 ) , ( 0 , 85 ) ; asymptotes: y = 2 9 x , y = 2 9 x

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( x 1 ) 2 9 ( y 2 ) 2 16 = 1

( x 1 ) 2 3 2 ( y 2 ) 2 4 2 = 1 ; vertices: ( 4 , 2 ) , ( 2 , 2 ) ; foci: ( 6 , 2 ) , ( 4 , 2 ) ; asymptotes: y = 4 3 ( x 1 ) + 2 , y = 4 3 ( x 1 ) + 2

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Questions & Answers

sin theta=3/4.prove that sec square theta barabar 1 + tan square theta by cosec square theta minus cos square theta
Umesh Reply
I want to know trigonometry but I can't understand it anyone who can help
Siyabonga Reply
Yh
Idowu
which part of trig?
Nyemba
functions
Siyabonga
trigonometry
Ganapathi
differentiation doubhts
Ganapathi
hi
Ganapathi
hello
Brittany
Prove that 4sin50-3tan 50=1
Sudip Reply
f(x)= 1 x    f(x)=1x  is shifted down 4 units and to the right 3 units.
Sebit Reply
f (x) = −3x + 5 and g (x) = x − 5 /−3
Sebit
what are real numbers
Marty Reply
I want to know partial fraction Decomposition.
Adama Reply
classes of function in mathematics
Yazidu Reply
divide y2_8y2+5y2/y2
Sumanth Reply
wish i knew calculus to understand what's going on 🙂
Dashawn Reply
@dashawn ... in simple terms, a derivative is the tangent line of the function. which gives the rate of change at that instant. to calculate. given f(x)==ax^n. then f'(x)=n*ax^n-1 . hope that help.
Christopher
thanks bro
Dashawn
maybe when i start calculus in a few months i won't be that lost 😎
Dashawn
what's the derivative of 4x^6
Axmed Reply
24x^5
James
10x
Axmed
24X^5
Taieb
Thanks for this helpfull app
Axmed Reply
secA+tanA=2√5,sinA=?
richa Reply
tan2a+tan2a=√3
Rahulkumar
classes of function
Yazidu
if sinx°=sin@, then @ is - ?
NAVJIT Reply
the value of tan15°•tan20°•tan70°•tan75° -
NAVJIT
0.037 than find sin and tan?
Jon Reply
cos24/25 then find sin and tan
Deepak Reply
Practice Key Terms 4

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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