A balloon in the shape of a sphere has volume V and radius r. Air is pumped into the balloon at a constant rate of 40π starting when time t = 0 and r = 0. At the same time, air begins to flow out of the balloon at a rate of 0.8πr. The balloon remains a sphere at all times.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2024). The mark or mark type is shown in brackets.

1. [B1] Obtain dV/dt = 40π - 0.8πr or equivalent
2. [B1] Obtain dV/dr = 4πr² or equivalent e.g. dV/dr = 4πr² dt/dt
3. [B1] Use the chain rule to obtain given answer (including the derivative)
4. [M1] Commence division and reach quotient of the form -5r ± 250 or 5r² = (50 – r)(Ar + B) + C and reach A = –5 and B = ± 250
5. [A1] Obtain quotient -5r – 250
6. [A1] Obtain remainder 12500
7. [B1ft] Prepare to integrate e.g. separate variables correctly
8. [DB1] Obtain term t
9. [M1] Obtain terms -A/2 r² + Br – Cln(50-r)
10. [A1ft] Obtain terms -5/2 r² - 250r - 12500ln(50-r)
11. [M1] Use t = 0, r = 0 to evaluate a constant or as limits in a solution containing terms of the form r², r, ln(50 – r) and t
12. [A1] Obtain final answer t = -5/2 r² – 250r – 12500ln(50-r) + 12500ln50
13. [B1] Obtain t = 70.5