Mathematics 9709 · AS & A Level · Differential equations
Differential equations — practice question
Liquid is entering a small tank that has a leak. At the beginning the tank contains no liquid, and after $t$ minutes the liquid volume in the tank is $V$ cm$^3$. The inflow rate is constant at $80$ cm$^3$ per minute. As a result of the leak, liquid is leaving the tank at a rate that, at any instant, is $kV$ cm$^3$ per minute, where $k$ is a positive constant.
(a)[7]
Write down a differential equation for the situation and solve it to demonstrate that $V = \frac{1}{k}(80 - 80e^{-kt})$.
(b)[3]
It is found that $V = 500$ when $t = 15$, so $k$ must satisfy $k = \frac{4 - 4e^{-15k}}{25}$. Use an iterative method based on this equation to determine $k$ correct to 2 significant figures. Begin with $k = 0.1$ and show the outcome of each iteration to 4 significant figures.
(c)[2]
Find how much liquid is in the tank 20 minutes after the liquid begins to flow, and state what happens to the tank's volume after a long time.
Worked solution & mark scheme
This 12-mark question has a full step-by-step worked solution and mark scheme. One marking point: “State $\frac{dV}{dt}=80-kV$” …