4 A watermelon is assumed to be spherical in shape while it is growing. Its mass, \(M \mathrm {~kg}\), and radius, \(r \mathrm {~cm}\), are related by the formula \(M = k r ^ { 3 }\), where \(k\) is a constant. It is also assumed that the radius is increasing at a constant rate of 0.1 centimetres per day. On a particular day the radius is 10 cm and the mass is 3.2 kg . Find the value of \(k\) and the rate at which the mass is increasing on this day.

## Question 4:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $1000k = 3.2 \Rightarrow k = \frac{3.2}{1000}$ or $\frac{2}{625}$ or $0.0032$ oe | M1A1 | |
| $\left(\frac{dM}{dr}\right) = 3kr^2$ | B1 | |
| $\frac{dM}{dt} = \frac{dM}{dr} \times \frac{dr}{dt}$ used e.g. $3 \times k \times 10^2 \times 0.1$ | M1 | Must eventually make $dM/dt$ subject cao. Non-calculus methods (e.g. $\to 0.09696$) can score only 1st 2 marks |
| $0.096$ | A1 [5] | |

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4 A watermelon is assumed to be spherical in shape while it is growing. Its mass, $M \mathrm {~kg}$, and radius, $r \mathrm {~cm}$, are related by the formula $M = k r ^ { 3 }$, where $k$ is a constant. It is also assumed that the radius is increasing at a constant rate of 0.1 centimetres per day. On a particular day the radius is 10 cm and the mass is 3.2 kg . Find the value of $k$ and the rate at which the mass is increasing on this day.

\hfill \mbox{\textit{CAIE P1 2012 Q4 [5]}}