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: 27

Answer

The larger gear rotates through an angle of $156^{\circ}$

Work Step by Step

We can convert $300^{\circ}$ to radians: $(300^{\circ})(\frac{\pi~rad}{180^{\circ}}) = 5.236~radians$ When the gears rotate, they will rotate through the same arc length. The radius of the smaller gear is 3.7 cm. We can find the arc length $d$ of the smaller gear when it rotates through an angle of $5.236~rad$: $d = (5.236~rad)(3.7~cm)$ $d = 19.373~cm$ The larger gear will rotate along this same arc length. The radius of the larger gear is 7.1 cm. We can find the angle $x$ through which the larger gear rotates: $x = \frac{19.373~cm}{7.1~cm}$ $x = 2.73~rad$ We can convert $2.73~rad$ to degrees: $(2.73~rad)(\frac{180^{\circ}}{\pi~rad}) = 156^{\circ}$ The larger gear rotates through an angle of $156^{\circ}$
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