Mathematics 9709 · AS & A Level · Differential equations
Differential equations — practice question
A water-filled tank is shaped as a cone with vertex $C$. Its axis is vertical, and the semi-vertical angle is $60^\circ$, as the diagram shows. When $t = 0$, the tank is completely full and the water depth is $H$. At that moment, a tap at $C$ is turned on and water starts draining out. The volume of water in the tank falls at a rate proportional to $\sqrt{h}$, where $h$ denotes the water depth at time $t$. The tank is empty when $t = 60$.
(i)[4]
Demonstrate that $h$ and $t$ obey a differential equation of the form $\frac{dh}{dt} = -Ah^{-\frac{3}{2}}$, where $A$ is a positive constant.
(ii)[6]
Solve the differential equation from part (i) and obtain an expression for $t$ in terms of $h$ and $H$.
(iii)[1]
Determine the time when the depth becomes $\frac{1}{2}H$.
Worked solution & mark scheme
This 11-mark question has a full step-by-step worked solution and mark scheme. One marking point: “State, or make clear, that $V=\pi h^3$” …