The diagram shows a semicircle $ABCD$ with radius $4$ cm and centre $O$. $ABCD$ is a trapezi

Question

The diagram shows a semicircle $ABCD$ with radius $4$ cm and centre $O$. $ABCD$ is a trapezium with height $y$ cm, $BC = 2x$ cm, $\angle COD = \theta$ radians and $AD$ is parallel to $BC$. Express $x$ and $y$ in terms of $\theta$, and show that the shaded area, $A$ cm$^2$, is given by $A = 8\pi - 16 \sin \theta(\cos \theta + 1)$. Find the value of $\theta$ which gives $A$ a stationary value. Find the stationary value of $A$ and show that this value of $A$ is a minimum.

Accepted Answer

(a) $x=4\cos\theta$, $y=4\sin\theta$, $A=8\pi-16\sin\theta(\cos\theta+1)$ (b) $\theta=\frac\pi3\text{ rad}$, $A_{\min}=8\pi-12\sqrt3\text{ cm}^2$

Differentiation - Problems Involving Maximisation and Minimisation

Differentiation - Problems Involving Maximisation and Minimisation

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