C4 January 2010 Q6
6. The area \(A\) of a circle is increasing at a constant rate of \(1.5\ \text{cm}^2\,\text{s}^{-1}\). Find, to 3 significant figures, the rate at which the radius \(r\) of the circle is increasing when the area of the circle is \(2\ \text{cm}^2\). (5)
| Scheme | Marks |
|---|---|
| \(\dfrac{\mathrm{d}A}{\mathrm{d}t} = 1.5\) | B1 |
| \(A = \pi r^2 \Rightarrow \dfrac{\mathrm{d}A}{\mathrm{d}r} = 2\pi r\) | B1 |
| When \(A = 2\) \(2 = \pi r^2 \Rightarrow r = \sqrt{\dfrac{2}{\pi}}\quad (= 0.797\,884\ \ldots)\) | M1 |
| \(\dfrac{\mathrm{d}A}{\mathrm{d}t} = \dfrac{\mathrm{d}A}{\mathrm{d}r} \times \dfrac{\mathrm{d}r}{\mathrm{d}t}\) \(1.5 = 2\pi r\dfrac{\mathrm{d}r}{\mathrm{d}t}\) | M1 |
| \(\dfrac{\mathrm{d}r}{\mathrm{d}t} = \dfrac{1.5}{2\pi\sqrt{\frac{2}{\pi}}} \approx 0.299\) awrt 0.299 | A1 |
| (5 marks) |