Question 3(c) Merit

Reading differentiability, stationary points and a limit straight off a drawn graph, then a log quotient, a related-rates sphere, a parametric gradient, and a rectangle inscribed in a curve.

A solid spherical lump of ice is melting while maintaining its spherical shape. At the instant when the radius is 6 cm, the radius is decreasing by 0.05 cm s⁻¹.

Find the rate at which the volume of the spherical ice ball is decreasing at the instant when the radius is 6 cm.

The 2025 paper, page 10 ↗ Marking schedule, page 5 ↗

Work it out

Try it on paper first. Stuck? Take one hint at a time.

1. 1 Which formula links V and r? What is dV/dr?

   V = (4/3)π r³, so dV/dr = 4π r².

2. 2 What rate does the question give you? Watch its sign.

   The radius is shrinking, so dr/dt = −0.05 cm s⁻¹.

3. 3 Chain the two together at r = 6.

   dV/dt = dV/dr × dr/dt = 4π (6)² × (−0.05) = −7.2π ≈ −22.62

What the marker wanted

• Achieved Both derivatives written down: dV/dr = 4π r² and dr/dt = −0.05.
• Merit dV/dt = −22.62, from those derivatives. Units are not needed, and leaving out the minus sign is not penalised.

In our words, from NZQA’s 2025 marking schedule.

How did it go?

Got it moves it further out. Review brings it back tomorrow.