Trigonometry (10th Edition)

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

Chapter 3 - Radian Measure and the Unit Circle - Section 3.2 Applications of Radian Measure - 3.2 Exercises - Page 105: 35

Answer

about 146 in.

Work Step by Step

Arc length s (for central angle $\theta$):$ \quad s=r\theta$, where $\theta$ is in radians Converting between Degrees and Radians 1. Multiply a degree measure by $\displaystyle \frac{\pi}{180}$ radian and simplify to convert to radians. 2. Multiply a radian measure by $\displaystyle \frac{180^{\mathrm{o}}}{\pi}$ and simplify to convert to degrees. ------------------- A pedal rotation of $\theta=180^{o}$ converted to radians: $\displaystyle \theta=180(\frac{\pi}{180}$ radian $)=\pi$ radians. The arc length of the pedal is $ s=r\theta=4.72\pi$. The chain moves this same distance, $ 4.72\pi$ in. The small gear rotates with this arc length through an angle $\theta$...(solve for $\theta$) . $ s=r\theta$ $\displaystyle \theta=\frac{s}{r}$ $\displaystyle \theta=\frac{4.72\pi}{1.38}\approx 3.42\pi$ The wheel, of which the small gear is a part of, rotates through this same angle of $ 3.42\pi$ radians. So, the arc the wheel will cover is . $s=r\theta\Rightarrow r=13.6(3.42\pi)\approx 146.12$ The bicycle will move about 146 in.
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