The diagram shows a quadrilateral divided into two right-angled triangles and a rectangle. Given tha

Question

The diagram shows a quadrilateral divided into two right-angled triangles and a rectangle.
Given that $\angle ADC = \theta{}^\circ$, $AB = 7$ cm, $AD = 15$ cm and $\angle BAD = \angle BCD = \angle AFB = 90{}^\circ$.

Find the length of $BC$ in the form of $a \sin \theta - b \cos \theta$. Hence show that the perimeter, $P$ cm, of the quadrilateral $ABCD$ is given by $P = 22 + 22 \sin \theta + 8 \cos \theta$.
Expressing $P$ in the form $P = 22 + R \sin(\theta + \alpha)$.
Hence state the maximum value of the perimeter and state the corresponding value of $\theta$.

Accepted Answer

(a) $BC=15\sin\theta-7\cos\theta$, so $P=22+22\sin\theta+8\cos\theta$ (b) $P=22+2\sqrt{137}\sin(\theta+19.983^\circ)$ (c) $P_{\max}=22+2\sqrt{137}\text{ cm}=45.4\text{ cm}$ when $\theta=70.0^\circ$

Trigonometry - R-Formula and Trigonometric Extrema

Trigonometry - R-Formula and Trigonometric Extrema

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