The figure shows a conical container with slant height $3$ cm and semi-vertical angle $\theta$.

Question

The figure shows a conical container with slant height $3$ cm and semi-vertical angle $\theta$. The container has a volume of $V$ cm$^3$ and $\theta$ varies so that $0 < \theta < \frac{\pi}{2}$. Show that $V = 9\pi \sin^2 \theta \cos \theta$. Show that $\frac{\mathrm{d}V}{\mathrm{d}\theta} = 9\pi [2 \sin \theta - 3 \sin^3 \theta]$. Find $\frac{\mathrm{d}^2 V}{\mathrm{d}\theta^2}$. Determine the values of $\sin \theta$ and $\cos \theta$ for which $V$ has a stationary value, leaving your answers in surd form. Find the maximum value of $V$ in terms of $\pi$.

Accepted Answer

(a) $V=9\pi\sin^2\theta\cos\theta$ (b) $\frac{\mathrm dV}{\mathrm d\theta}=9\pi(2\sin\theta-3\sin^3\theta)$ (c) $\frac{\mathrm d^2V}{\mathrm d\theta^2}=9\pi(2\cos\theta-9\sin^2\theta\cos\theta)$ (d) $\sin\theta=\frac{\sqrt6}{3}$, $\cos\theta=\frac{\sqrt3}{3}$ (e) $V_{\max}=2\sqrt3\pi\text{ cm}^3$

Differentiation - Problems Involving Maximisation and Minimisation

Differentiation - Problems Involving Maximisation and Minimisation

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