Specimen 2017 H2 Mathematics Paper 1 Question 1

Differentiation II: Maxima, Minima, Rates of Change

Answers

dCdt=0.7854 cm/s,{\frac{\mathrm{d}C}{\mathrm{d}t} = 0.7854 \textrm{ cm/s},}
dAdt=1.1781 cm2/s.{\frac{\mathrm{d}A}{\mathrm{d}t} = 1.1781 \textrm{ cm}^2\textrm{/s}.}

Full solutions

Let C{C} and A{A} denote the circumference and area of the circle respectively.
C=πDdCdD=π\begin{align*} C &= \pi D \\ \frac{\mathrm{d}C}{\mathrm{d}D} &= \pi \end{align*}
dCdt=dCdD×dDdt=π×0.25=0.7854 cm/s  ■\begin{align*} \frac{\mathrm{d}C}{\mathrm{d}t} &= \frac{\mathrm{d}C}{\mathrm{d}D} \times \frac{\mathrm{d}D}{\mathrm{d}t} \\ &= \pi \times 0.25 \\ &= 0.7854 \textrm{ cm/s} \; \blacksquare \end{align*}
A=πr2=πD24dAdD=πD2\begin{align*} A &= \pi r^2 \\ &= \frac{\pi D^2}{4} \\ \frac{\mathrm{d}A}{\mathrm{d}D} &= \frac{\pi D}{2} \end{align*}
dAdt=dAdD×dDdt=π×1.5×22×0.25=1.1781 cm2/s  ■\begin{align*} \frac{\mathrm{d}A}{\mathrm{d}t} &= \frac{\mathrm{d}A}{\mathrm{d}D} \times \frac{\mathrm{d}D}{\mathrm{d}t} \\ &= \frac{\pi \times 1.5 \times 2}{2} \times 0.25 \\ &= 1.1781 \textrm{ cm}^2\textrm{/s} \; \blacksquare \end{align*}