Final answer:
To find the width of the tower at a height of 151 feet, you need to determine the equation of the hyperbola that represents the sides of the cooling tower. Using the given information, you can find the values of a and b in the hyperbola equation and then substitute the height to find the width.
Step-by-step explanation:
To find the width of the tower at a height of 151 feet, we need to determine the equation of the hyperbola that represents the sides of the cooling tower.
The general equation of a hyperbola with vertical transverse axis is:
(y - k)^2 / a^2 - (x - h)^2 / b^2 = 1
Given the diameter at the bottom of the tower is 274 feet, we know that the center of the hyperbola is at (0, 0) and the value of b is half of the diameter, which is 137 feet.
Using the point (0, 201) where the tower is 401 feet above the ground, we can substitute the values into the equation to find a:
(201 - 0)^2 / a^2 - (0 - 0)^2 / 137^2 = 1
a = √((201 - 0)^2 - (0 - 0)^2 / 137^2) = √((201)^2 - (137)^2) = √(40401 - 18769) = √21632 ≈ 147.1
Now, using the point (0, 151), we can find the width at that height:
(151 - 0)^2 / (147.1)^2 - (x - 0)^2 / (137)^2 = 1
151^2 / (147.1)^2 = (x - 0)^2 / (137)^2
x^2 = 151^2 * (137)^2 / (147.1)^2
x = √((151)^2 * (137)^2 / (147.1)^2) = 119.6
Therefore, the width of the tower at a height of 151 feet is approximately 119.6 feet.