Transcribed Image Text: The sides of a nuclear power plant cooling tower form a hyperbola. The diameter of the bottom of the towe is 286 feet. The smallest diameter of the tower is 168 which is 435 feet above the ground. The tower is 590 feet tall. 168 ft 590 ft 435 ft 286 ft Photo remixed from work by Jiří Sedláček r?!, cC-BY-SA Find the width of the tower at a height of 128 feet, to 1 decimal place. feet.
To find the width of the tower at a height of 128 feet, we can use the equation of a hyperbola. The equation of a hyperbola in standard form is:
(x^2 / a^2) – (y^2 / b^2) = 1
Where “a” is the distance from the center to the vertex along the x-axis, and “b” is the distance from the center to the vertex along the y-axis.
In this case, the tower is tilted, so we need to rotate the hyperbola equation. Since the tower has a vertical major axis, the equation becomes:
(y^2 / a^2) – (x^2 / b^2) = 1
In our problem, the smallest diameter of the tower is 168 feet, which corresponds to “2a.” Therefore, we can find “a” by dividing 168 by 2, giving us a value of 84 feet.
The tower is 435 feet above the ground, so this corresponds to the value of “b.”
Now, we need to find the value of “x” when the tower is at a height of 128 feet.
To solve for the width at a particular height, we can substitute the values into the hyperbola equation and solve for “x.”
(128^2 / a^2) – (x^2 / b^2) = 1
Plugging in the values of “a” and “b” from earlier, we get:
(128^2 / 84^2) – (x^2 / 435^2) = 1
Simplifying the equation, we have:
(128^2 / 84^2) – (x^2 / 435^2) = 1
(16384 / 7056) – (x^2 / 189225) = 1
2.32 – (x^2 / 189225) = 1
Now, let’s solve for “x.” Rearranging the equation, we have:
(x^2 / 189225) = 2.32 – 1
(x^2 / 189225) = 1.32
Cross-multiplying, we get:
x^2 = 1.32 * 189225
x^2 = 249813
Taking the square root of both sides, we find:
x = sqrt(249813)
Now, we can calculate the width of the tower at a height of 128 feet by substituting this value into the equation. To one decimal place, we have:
x = sqrt(249813)
x ≈ 499.8 feet
Therefore, the width of the tower at a height of 128 feet is approximately 499.8 feet.