Hyperbolas, find width of cooling tower.....

Question

The sides of a nuclear power plant cooling tower form a hyperbola. The diameter of the bottom of the tower is 291 feet. The smallest diameter of the tower is 182 feet which is 419 feet above the ground. The tower is 566 feet tall.

Find the width of the tower at a height of 154 feet.

Answer 1

Let center at hyperbola at origin, x-axis horizontal left, y - axis vertical up. Then we have such points on hyperbola: (182/2, 0) = (91, 0); ( 291/2, - 419);

Equation of hyperbola: x²/a² - y²/b² = 1; 91²/a² = 1; a = 91; 291²/(4·91²) - 419²/b² = 1; b = 335.8471

x²/91² - y²/335.8471² = 1

If h = 154, yhen y = - (419 - 154) = - 265

x²/91² - 265²/335.8471² =1; x = 115.9 feet

Width: 2x = 231.8 feet