A cafe makes x litres of coffee each morning. The cafe's profit each morning, C, measured in dollars, is modelled by the following equation $C = \frac{x}{10}(k^2 - \frac{3}{100}x^2)$ where k is a positive constant.

attempt to expand given expression OR attempt at product rule $C=\frac{x k^{2}}{10}-\frac{3 x^{3}}{1000}$ $\frac{\mathrm{d} C}{\mathrm{d} x}=\frac{k^{2}}{10}-\frac{9 x^{2}}{1000}$ Note: Award $M1$ for power rule correctly applied to at least one term and $A1$ for correct answer.

equating their $\frac{\mathrm{d} C}{\mathrm{d} x}$ to zero $\frac{k^{2}}{10}-\frac{9 x^{2}}{1000}=0$ $x^{2}=\frac{100 k^{2}}{9}$ $x=\frac{10 k}{3}$ substituting their $x$ back into given expression (M1) $C_{\max }=\frac{10 k}{30}\left(k^{2}-\frac{300 k^{2}}{900}\right)$ $C_{\max }=\frac{2 k^{3}}{9}\left(0.222 \ldots k^{3}\right)$

substituting 20 into given expression and equating to 426

$426=\frac{20}{10}\left(k^{2}-\frac{3}{100}(20)^{2}\right)$ $k=15$ A1