The diagram shows an inverted cone with slant height $3$ cm and semi-vertical angle $\theta$ rad

Question

The diagram shows an inverted cone with slant height $3$ cm and semi-vertical angle $\theta$ radians. State the range of values of $\theta$. Show that the volume of the cone, $V$ cm$^3$, is given by $V = 9\pi \sin^2 \theta \cos \theta$. Find the values of $\sin \theta$ and $\cos \theta$ for which $V$ has a stationary value. Hence find the stationary value of $V$ and determine its nature. Leave your answer in terms of $\pi$.

Accepted Answer

(i) $0<\theta<\dfrac{\pi}{2}$ (ii) $V=9\pi\sin^2\theta\cos\theta$ (iii) $\sin\theta=\dfrac{\sqrt6}{3},\ \cos\theta=\dfrac{\sqrt3}{3}$ (iv) $2\sqrt3\pi\text{ cm}^3$, maximum

Techniques of Differentiation - Derivatives of Trigonometric Functions

Techniques of Differentiation - Derivatives of Trigonometric Functions

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