SOLUTION: The sides of a nuclear power plant cooling tower form a hyperbola. The diameter of the bottom of the tower is 295 feet. The smallest diameter of the tower is 166 feet which is 407
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-> SOLUTION: The sides of a nuclear power plant cooling tower form a hyperbola. The diameter of the bottom of the tower is 295 feet. The smallest diameter of the tower is 166 feet which is 407
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Question 1143998: The sides of a nuclear power plant cooling tower form a hyperbola. The diameter of the bottom of the tower is 295 feet. The smallest diameter of the tower is 166 feet which is 407 feet above the ground. The tower is 524 feet tall.
Find the width of the tower at a height of 233 feet.
The numbers are not pleasant; you can crunch them on your calculator as easily as I can on mine. So I will just help you set up the problem and let you have the fun of doing the calculations.
The equation of a hyperbola is simplest if the center of the hyperbola is at the origin. So let the origin be the midpoint of the diameter of the tower at its narrowest point.
Then, since the smallest diameter of the tower is 166 feet, one vertex of the hyperbola is at (83,0). At this point, we know the equation of the hyperbola is
To find b and complete the equation of the hyperbola, use another known point on the hyperbola.
The other known point is where a branch of the hyperbola touches the ground. Since the diameter of the base of the tower is 295 feet and the narrowest point of the tower is 407 feet above the ground, the coordinates of this other known point are (147.5,-407).
Then, to finish the problem...
(1) plug those coordinates into the equation for the hyperbola to determine b^2; and
(2) Use the completed equation of the hyperbola with y=-174 (corresponding to 233 feet above the ground) to solve for x.
Then remember that the x you got is the RADIUS of the tower at a height of 233 feet; the problem asks for the width (diameter) of the tower at that height.
