Trigonometry (10th Edition)

Published by Pearson
ISBN 10: 0321671775
ISBN 13: 978-0-32167-177-6

Chapter 6 - Inverse Circular Functions and Trigonometric Equations - Section 6.3 Trigonometric Equations II - 6.3 Exercises - Page 275: 52a

Answer

In October, the average monthly temperature in Phoenix is $70.5^\circ F$.

Work Step by Step

$$f(x)=19.5\cos[\frac{\pi}{6}(x-7)]+70.5$$ The question asks when the average monthly temperature in Phoenix is $70.5^\circ F$ In other words, find $x$ so that $f(x)=70.5$. Therefore, we can replace $f(x)=70.5$ to the given equation and find $x$: $$19.5\cos[\frac{\pi}{6}(x-7)]+70.5=70.5$$ $$19.5\cos[\frac{\pi}{6}(x-7)]=0$$ $$\cos[\frac{\pi}{6}(x-7)]=0$$ $$\frac{\pi}{6}(x-7)=\cos^{-1}0=\frac{\pi}{2}$$ $$x-7=\frac{\pi}{2}\times\frac{6}{\pi}$$ $$x-7=3$$ $$x=10$$ $x=10$ refers to the month October. Thus, in October, the average monthly temperature in Phoenix is $70.5^\circ F$.
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