Water is pouring into a conical cistern at the rate of 18 cubic feet per minute. If the height of the cistern is 21 feet and the radius of its circular opening is 7 feet, how fast is the water level rising when the water is 16 feet deep? Round answer to nearest hundredth.
A) 0.12 ft/min.
B) 0.54 ft/min.
C) 0.20 ft/min.
D) 1.49 ft/min.

Question

Quizplus · Accepted Answer

The rate at which the water level rises can be found using the formula for the volume of a cone $V = \frac{1}{3}\pi r^2 h$ and the given rate of volume increase. Given the dimensions of the cistern and the depth of the water, we can use similar triangles to find the radius of the water surface when the depth is 16 feet. The ratio of the radius to the height in the cistern is constant, so $r:h = 7:21$, which simplifies to $r = \frac{1}{3}h$. When $h = 16$, $r = \frac{16}{3}$. Using the given rate of volume increase ($dV/dt = 18$ cubic feet per minute) and the formula for the volume of a cone, we differentiate both sides with respect to time to find $dh/dt$, the rate at which the water level rises. Solving this equation with the given values leads to $dh/dt = 0.20$ ft/min when the water is 16 feet deep.

Water Is Pouring into a Conical Cistern at the Rate