Question 1185825
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.