The diagram shows a semicircle construction $ABCD$ with radius $r$ cm and centre $O$. The supp

Question

The diagram shows a semicircle construction $ABCD$ with radius $r$ cm and centre $O$. The supporting structure consists of trapezium $ABCD$, radii $OB$ and $OC$. Trapezium $ABCD$ is such that $BC$ is parallel to $AD$ and has a height of $h$ cm. Let angle $ABO = \theta$ radians and $BC = 2x$ cm. State the range of values of $\theta$. Express $x$ and $h$ in terms of $\theta$ and $r$. Hence show that the shaded area, $A$ cm$^2$, is given by $A = \frac{1}{2} r^2 (\pi - 2 \sin 2\theta + \sin 4\theta).$ Given that $\theta$ can vary, find the value of $\theta$ for which $A$ has a stationary value. Hence find the stationary value of $A$ in terms of $r$ and $\pi$, and determine its nature.

Accepted Answer

(i) $\dfrac{\pi}{4}<\theta<\dfrac{\pi}{2}$ (ii) $x=-r\cos2\theta,\ h=r\sin2\theta$ (iii) $A=\dfrac12r^2(\pi-2\sin2\theta+\sin4\theta)$ (iv) $\theta=\dfrac{\pi}{3}$ (v) $A=\dfrac{r^2}{4}(2\pi-3\sqrt3)\text{ cm}^2$, minimum

Techniques of Differentiation - Derivatives of Trigonometric Functions

Techniques of Differentiation - Derivatives of Trigonometric Functions

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