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Question 1185825: A large cooling tower in the shape of a hyperboloid is used to remove the heat from a nuclear power plant. The shortest width of the cooling tower is 56 meters. From this part, the length to the top of the cooling tower is 65 meters. The diameter of the top of the cooling tower is 60 meters. What is the equation of the hyperbola that represents the sides of the cooling tower? Set the middle of the shortest width as the origin.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to find the equation of the hyperbola:
1. **Orientation:** Since the shortest width is horizontal, the hyperbola's transverse axis is horizontal. The general equation for such a hyperbola centered at the origin is:
```
(x^2 / a^2) - (y^2 / b^2) = 1
```
2. **Finding 'a':** The shortest width is the distance between the vertices of the hyperbola, which is 2*a. We're given that this width is 56 meters. Therefore:
```
2a = 56
a = 28
a^2 = 784
```
3. **Finding 'b':** We need to use the information about the top of the tower to find 'b'. We know the following:
* The diameter of the top is 60 meters, so the radius is 30 meters. This means when y = 65, x = 30.
* The center of the hyperbola is at the middle of the shortest width which is also the origin.
4. **Substitute and solve for b^2:** Plug the values of x, y, and a^2 into the hyperbola equation:
```
(30^2 / 784) - (65^2 / b^2) = 1
(900 / 784) - (4225 / b^2) = 1
1.148 - (4225 / b^2) = 1
0.148 = 4225 / b^2
b^2 = 4225 / 0.148
b^2 ≈ 28547.3
```
5. **Final Equation:** Substitute the values of a^2 and b^2 back into the hyperbola equation:
```
(x^2 / 784) - (y^2 / 28547.3) = 1
```
Therefore, the equation of the hyperbola that represents the sides of the cooling tower is approximately (x^2 / 784) - (y^2 / 28547.3) = 1.
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