Paint is poured into a tray where it forms a circular pool with a uniform thickness of 0.5 cm. If the paint is poured at a constant rate of \(4{\text{ c}}{{\text{m}}^3}{{\text{s}}^{ - 1}}\), find the rate of increase of the radius of the circle when the radius is 20 cm.

\(V = 0.5\pi {r^2}\)     (A1)

EITHER

\(\frac{{dV}}{{dr}} = \pi r\)     A1

\(\frac{{dV}}{{dt}} = 4\)     (A1)

applying chain rule     M1

for example \(\frac{{dr}}{{dt}} = \frac{{dV}}{{dt}} \times \frac{{dr}}{{dV}}\)

OR

\(\frac{{dV}}{{dt}} = \pi r\frac{{dr}}{{dt}}\)     M1A1

\(\frac{{dV}}{{dt}} = 4\)     (A1)

THEN

\(\frac{{dr}}{{dt}} = 4 \times \frac{1}{{\pi r}}\)     A1

when \(r = 20,{\text{ }}\frac{{dr}}{{dt}} = \frac{4}{{20\pi }}{\text{ or }}\frac{1}{{5\pi }}{\text{ (cm}}\,{{\text{s}}^{ - 1}})\)     A1

Note: Allow h instead of 0.5 up until the final A1.

[6 marks]

There was a large variety of methods used in this question, with most candidates choosing to implicitly differentiate the expression for volume in terms of r.