Mathematics 9709 · AS & A Level · Differential equations
Differential equations — practice question
A spherical balloon has volume $V$ and radius $r$. Air is pumped into the balloon at a steady rate of $40\pi t$ from the moment $t = 0$ when $r = 0$. At the same time, air also escapes from the balloon at $0.8\pi r$. The balloon stays spherical throughout.
(a)[3]
Show that $r$ and $t$ satisfy the differential equation $\frac{dr}{dt} = \frac{50 - r}{5r^2}$.
(b)[3]
Find the quotient and remainder when $5r^2$ is divided by $50 - r$.
(c)[6]
Solve the differential equation from part (a), producing an expression for $t$ in terms of $r$.
(d)[1]
Find the value of $t$ when the balloon’s radius is $12$.
Worked solution & mark scheme
This 13-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Obtain $\frac{dV}{dt}=40\pi-0.8\pi r$ or a correct equivalent” …