Final Answer:
The width of the tower at a height of 72 feet is approximately 247 feet.
Step-by-step explanation:
Define the given information:
Bottom diameter (d_b) = 314 feet
Smallest diameter (d_s) = 206 feet
Height of smallest diameter (h_s) = 393 feet
Total height (h_t) = 578 feet
Desired height (h_d) = 72 feet
Formulate the equation for the hyperbola:
We can use the standard equation for a horizontal hyperbola with center at (0,0):
(y-0)^2 / a^2 - (x-0)^2 / b^2 = 1
where:
a is the vertical distance from the center to a focus
b is the horizontal distance from the center to a vertex
Relate the given information to the equation:
The bottom diameter corresponds to the width of the tower at ground level (y = 0). Therefore, the bottom diameter is equal to 2b.
The smallest diameter corresponds to the width of the tower at a height of 393 feet (y = 393). Therefore, the smallest diameter is equal to 2b - 2a.
Solve for the parameters:
Substitute the given values:
2b = 314 feet
2b - 2a = 206 feet
Solve for a and b:
a = 54 feet
b = 157 feet
Find the width at the desired height:
Substitute the desired height (h_d = 72 feet) into the equation:
(72-0)^2 / 54^2 - (x-0)^2 / 157^2 = 1
Solve for x:
x ≈ 126 feet
Therefore, the width of the tower at a height of 72 feet is approximately 126 feet * 2 = 247 feet.