Find the width of the tower at a height of 14 feet, to 1 decimal place.

Question

The sides of a nuclear power plant cooling tower form a hyperbola. The diameter of the bottom of the tower is 306 feet. The smallest diameter of the tower is 176 which is 390 feet above the ground. The tower is 564 feet tall.

Find the width of the tower at a height of 14 feet, to 1 decimal place.

Answer 1

To make the math easier, let's set the problem up so the x-axis runs up the center of the tower with the ground at x=0, and the y-value will be the distance from the center--which will be half the width of the tower.

Doing that gives two points or the hyperbola: (0,153) and (390, 88). The second point being the narrowest part of the tower means it's also the vertex of the parabola which means the hyperbola's center is at (390, 0). Now we can take the equation of a hyperbola in standard form and use the two known points to find all the constants. At the end, we'll use that equation to find the value at 14 feet.

(x-h)²/a² - (y-k)²/b² = 1

h and k are the coordinates of the center, so putting in the values for the narrowest part gives

(390-390)²/a² - (88-0)²/b² =1

0²/a² - 88²/b² = 1

88²/b² = 1

so b²=88²=7744

We can now use the other point at the base to solve for a²

(0-390)²/a² - (156-0)²/7744 = 1

390²/a² - 156²/7744 = 1

152,100/a² - 3.142 = 1

152,100/a² = 4.142

a² = 152,100/4.142 = 36716

Now for the answer, we put in 14 for x and find y

(14-390)²/36716 - (y-0)²/7744 = 1

3.851 - y²/7744 = 1

-y²/7744 = -2.851

y² = 22074

y = 148.57

Remember that y is half the width of the tower, so at 14 feet off the ground, the tower is 297.1 feet wide.