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10.2 The hyperbola  (Page 7/13)

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Graph the hyperbola given by the standard form of an equation   ( y + 4 ) 2 100 − ( x − 3 ) 2 64 = 1.   Identify and label the center, vertices, co-vertices, foci, and asymptotes.

center:   ( 3 , − 4 ) ;   vertices:   ( 3 , − 14 )   and   ( 3 , 6 ) ;   co-vertices:   ( − 5 , − 4 ) ;   and   ( 11 , − 4 ) ;   foci:   ( 3 , − 4 − 2 41 )   and   ( 3 , − 4 + 2 41 ) ;   asymptotes:   y = ± 5 4 ( x − 3 ) − 4

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Solving applied problems involving hyperbolas

As we discussed at the beginning of this section, hyperbolas have real-world applications in many fields, such as astronomy, physics, engineering, and architecture. The design efficiency of hyperbolic cooling towers is particularly interesting. Cooling towers are used to transfer waste heat to the atmosphere and are often touted for their ability to generate power efficiently. Because of their hyperbolic form, these structures are able to withstand extreme winds while requiring less material than any other forms of their size and strength. See [link] . For example, a 500-foot tower can be made of a reinforced concrete shell only 6 or 8 inches wide!

Cooling towers at the Drax power station in North Yorkshire, United Kingdom (credit: Les Haines, Flickr)

The first hyperbolic towers were designed in 1914 and were 35 meters high. Today, the tallest cooling towers are in France, standing a remarkable 170 meters tall. In [link] we will use the design layout of a cooling tower to find a hyperbolic equation that models its sides.

Solving applied problems involving hyperbolas

The design layout of a cooling tower is shown in [link] . The tower stands 179.6 meters tall. The diameter of the top is 72 meters. At their closest, the sides of the tower are 60 meters apart.

Project design for a natural draft cooling tower

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.

We are assuming the center of the tower is at the origin, so we can use the standard form of a horizontal hyperbola centered at the origin:   x 2 a 2 − y 2 b 2 = 1 , where the branches of the hyperbola form the sides of the cooling tower. We must find the values of   a 2   and   b 2 to complete the model.

First, we find   a 2 .   Recall that the length of the transverse axis of a hyperbola is   2 a .   This length is represented by the distance where the sides are closest, which is given as   65.3   meters. So,   2 a = 60.   Therefore,   a = 30   and   a 2 = 900.

To solve for   b 2 , we need to substitute for   x   and   y   in our equation using a known point. To do this, we can use the dimensions of the tower to find some point   ( x , y )   that lies on the hyperbola. We will use the top right corner of the tower to represent that point. Since the y -axis bisects the tower, our x -value can be represented by the radius of the top, or 36 meters. The y -value is represented by the distance from the origin to the top, which is given as 79.6 meters. Therefore,

x 2 a 2 − y 2 b 2 = 1 Standard form of horizontal hyperbola .            b 2 = y 2 x 2 a 2 − 1 Isolate  b 2               = ( 79.6 ) 2 ( 36 ) 2 900 − 1 Substitute for  a 2 , x ,  and  y               ≈ 14400.3636 Round to four decimal places

The sides of the tower can be modeled by the hyperbolic equation

x 2 900 − y 2 14400.3636   = 1 , or     x 2 30 2 − y 2 120.0015 2   = 1
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Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
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Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
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field is a region of space under the influence of some physical properties
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Another formula for Acceleration
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a=v/t. a=f/m a
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Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
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specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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