A 200-liter tank, initially full of water, develops a leak at the bottom.

Question

A 200-liter tank, initially full of water, develops a leak at the bottom. Given that 20% of the water leaks out in the first 5 minutes, find the amount of water left in the tank t minutes after the leak develops: (a) if the water drains off at a rate proportional to the amount of water present. (b)  if the water drains off at a rate proportional to the product of the time elapsed and the amount of water present.

Answer 1

Let V(t) = volume of water in tank as function of time.

a) dV/dt= kV ... rate is proportional to volume ... rearrange to find dV/V=kdt and integrate both sides to find ln(V)=kt+C and V(t)= e^kt+C = e^Ce^kt. Note that V(0)=200 thus e^C=200L and that V(5)= 200*,8=160L =200e^5k, thus k= -.0446 min^-1. V(t)=200e^-.0446t.

b) dV/dt= ktV ... rate is proportional to volume*time. rearrange to find dV/V=ktdt and lnV= .5kt²+C. Find k & C using the initial conditions.