The rate at which water flows out of a drain in the bottom of a certain tank is proportional to the height of water in the tank. The tank is a vertical cylinder with cross-sectional area of 1.0 m², so that every 1 cm in height represents 10 L. If the flow is 10 L/min (i.e. 1 cm/min) when the water level is 700 cm, how long will it take for the level to go from 700 cm to 30 cm? A) 4586 minutes B) 670 minutes C) 9 minutes D) 2205 minutes

Question

Quizplus · Accepted Answer

Let H be the height of the water in cm and t be the time in minutes. Then, we have:
dH/dt = kH, where k is the constant of proportionality.
Also, we know that when H = 700, dH/dt = 10 L/min = 1 cm/min. Therefore,
k = dH/dtH = 1/700
We need to find t such that H = 30. Integrating both sides of the differential equation, we get:
ln(H/700) = kt
ln(30/700) = (1/700) t
t = 700 ln(30/700) ≈ 2205 minutes

The Rate at Which Water Flows Out of a Drain