Trigonometry (11th Edition) Clone

Published by Pearson
ISBN 10: 978-0-13-421743-7
ISBN 13: 978-0-13421-743-7

Chapter 3 - Radian Measure and the Unit Circle - Section 3.2 Applications of Radian Measure - 3.2 Exercises - Page 111: 33


about 18.7 cm

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. ---------------- For the smaller gear, the length of the arc that turns is $ s=r\theta$ $\theta=80^{o}$ needs to be converted to radians (case 1): $s=r\displaystyle \cdot 80\cdot\frac{\pi}{180}=11.7\cdot 4\cdot\frac{\pi}{9}=\frac{11.7\cdot 4\cdot\pi}{9}$ This arc length is the same as the arc length of the larger wheel, rotating through the angle of $50.0^{o}.$ $ s=r\theta\qquad$ ( ... solve for $r$) $\theta = 50.0\displaystyle \cdot\frac{\pi}{180}=\frac{5\pi}{18}$ $\displaystyle \frac{11.7\cdot 4\cdot\pi}{9}=r\cdot\frac{5\pi}{18}$ $ r=\displaystyle \frac{11.7\cdot 4\cdot\pi\cdot 18}{9\cdot 5\pi}=\frac{11.7\cdot 4\cdot 2}{5}\approx$18.72 The radius of the bigger wheel is about 18.7 cm
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