Water is flowing into a vertical cylindrical tank at the rate of 5 m³/min. If the radius of the tank is 3 m, then the rate at which the height of the water is rising is A) $\frac { 9 } { 5 \pi }$ B) $\frac { 3 } { 5 \pi }$ C) $\frac { 5 } { 6 \pi }$ D) $\frac { 5 } { 9 \pi }$ E) $\frac { 5 } { 3 \pi }$

Question

Quizplus · Accepted Answer

The volume of a cylinder is given by $ V = \pi r^2 h $. Differentiating with respect to time $ t $, we get $ \frac{dV}{dt} = \pi r^2 \frac{dh}{dt} $. Given $ \frac{dV}{dt} = 5 $ m$^3$/min and $ r = 3 $ m, we solve for $ \frac{dh}{dt} $: $ 5 = \pi (3)^2 \frac{dh}{dt} $, leading to $ \frac{dh}{dt} = \frac{5}{9\pi} $.

Water Is Flowing into a Vertical Cylindrical Tank at the Rate