Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 3 - Section 3.3 - Derivatives of Trigonometric Functions - 3.3 Exercises: 56

Answer

$$\lim\limits_{\theta\to0^+}\frac{A(\theta)}{B(\theta)}=0$$

Work Step by Step

1) First, find the area of the triangle $B(\theta)$ Here, we know the length of 2 triangle sides $(\overline{RP}=\overline{RQ}=10cm)$ and the included angle $(\angle R=\theta)$. Therefore, to find the area of the triangle, we would use this formula $$A=\frac{1}{2}ab\sin\angle C$$ $A$: the area of the triangle $a$ and $b$: the length of 2 sides of the triangle $\angle C$: the included angle of 2 sides mentioned above That means, here in $\triangle PRQ$, its area would be $$B(\theta)=\frac{1}{2}\overline{RQ}\overline{RP}\sin\angle R$$ $$B(\theta)=\frac{1}{2}\times10\times10\times\sin\theta$$ $$B(\theta)=50\sin\theta (cm^2)$$ 2) Find the area of semicircle $A(\theta)$ To find the area of the semicircle, we need to calculate $\overline{PQ}$. We can do that from the information given in $\triangle PRQ$ $\triangle PRQ$, first, is not a right triangle. Then, we already know the length of 2 sides and the included angle of these 2 sides. Therefore, the most appropriate way to find the length of the remaining side is to use Law of Cosine. In detail, $$c^2=a^2+b^2-2ab\cos\angle C$$ $a$, $b$ and $c$: the length of 3 sides of the triangle $\angle C$: the included angle of 2 sides with the length $a$ and $b$ respectively Applying to $\triangle PRQ$: $$\overline{PQ}^2=\overline{RQ}^2+\overline{RP}^2-2\overline{RQ}\overline{RP}\cos\angle C$$ $$\overline{PQ}^2=10^2+10^2-2\times10\times10\times\cos\theta$$ $$\overline{PQ}^2=200-200\cos\theta$$ $$\overline{PQ}^2=100\times2(1-\cos\theta)$$ $$\overline{PQ}=10\sqrt{2(1-\cos\theta)}$$ $PQ$ is the diameter of the semicircle. Therefore, the area of the semicircle would be $$A(\theta)=\frac{\pi(\overline{PQ}/2)^2}{2}$$ $$A(\theta)=\frac{\pi(5\sqrt{2(1-\cos\theta)})^2}{2}$$ $$A(\theta)=\frac{\pi\times25\times2(1-\cos\theta)}{2}$$ $$A(\theta)=25\pi(1-\cos\theta)(cm^2)$$ 3) Calculate $\lim\limits_{\theta\to0^+}\frac{A(\theta)}{B(\theta)}$ $$\lim\limits_{\theta\to0^+}\frac{A(\theta)}{B(\theta)}=\lim\limits_{\theta\to0^+}\frac{25\pi(1-\cos\theta)}{50\sin\theta}=\lim\limits_{\theta\to0^+}\frac{-\pi(\cos\theta-1)}{2\sin\theta}$$ $$=\frac{-\pi}{2}\lim\limits_{\theta\to0^+}\frac{\cos\theta-1}{\sin\theta}$$ Divide both numerator and denominator by $\theta$ $$=\frac{-\pi}{2}\lim\limits_{\theta\to0^+}\frac{\frac{\cos\theta-1}{\theta}}{\frac{\sin\theta}{\theta}}$$ $$=-\frac{\pi}{2}\times\frac{0}{1}=0$$
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