An inverted pyramid is being filled with water at a constant rate of 60 cubic centimeters per second. The pyramid, at the top, has the shape of a square with sides of length 8 cm, and the height is 12 cm.

Question

Find the rate at which the water level is rising when the water level is 8 cm.

Answer 1

the width (w) of the base is 2/3 of the height (h) ... w = 2 h / 3

the filled volume is ... 1/3 * h * w^2

substituting ... v = 1/3 * h * (2 h / 3)^2 = 4/27 * h^3

differentiating with respect to time ... dv/dt = 4/9 h^2 dh/dt

dh/dt = (9 dv/dt) / (4 h^2) = 9 * 60 / (4 * 8^2) ... cm/sec