Mathematics 9709 · AS & A Level · Differential equations
Differential equations — practice question
A sizeable cylindrical tank is being used to hold water. The tank has a circular base of radius $4$ metres. At time $t$ minutes, the water depth in the tank is $h$ metres. A tap is fitted at the bottom of the tank. When the tap is open, water leaves the tank at a rate proportional to the square root of the volume of water in the tank.
(a)[4]
Show that $\frac{dh}{dt} = -\lambda\sqrt{h}$, for some positive constant $\lambda$.
(b)[6]
At time $t = 0$ the tap is opened. It is stated that $h = 4$ when $t = 0$ and that $h = 2.25$ when $t = 20$. Solve the differential equation to find an expression for $t$ in terms of $h$, and hence determine how long the tank takes to empty.
Worked solution & mark scheme
This 10-mark question has a full step-by-step worked solution and mark scheme. One marking point: “State or imply $\dfrac{dV}{dt}=\pm k\sqrt V$ or else $\dfrac{dV}{dt}=16\pi\dfrac{dh}{dt}$.” …