Liquid is pouring into a container at a constant rate of \(20 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\) and is leaking out at a rate proportional to the volume of liquid already in the container.
Explain why, at time \(t\) seconds, the volume, \(V \mathrm {~cm} ^ { 3 }\), of liquid in the container satisfies the differential equation
$$\frac { \mathrm { d } V } { \mathrm {~d} t } = 20 - k V$$
where \(k\) is a positive constant.
The container is initially empty.
By solving the differential equation, show that
$$V = A + B \mathrm { e } ^ { - k t }$$
giving the values of \(A\) and \(B\) in terms of \(k\).
Given also that \(\frac { \mathrm { d } V } { \mathrm {~d} t } = 10\) when \(t = 5\),
find the volume of liquid in the container at 10 s after the start.