Elementary Technical Mathematics

Published by Brooks Cole
ISBN 10: 1285199197
ISBN 13: 978-1-28519-919-1

Chapter 12 - Section 12.6 - Radian Measure - Exercise - Page 418: 31

Answer

(a.) $26.2\; m$ (b.) $327 \;m^2$ (c.) $56 \;m^2$.

Work Step by Step

Given values are:- Radius $r=25.0\; m$. Central angle $\theta = 60.0^{\circ}$. All sides of the triangle are equal. (equilateral triangle) Therefore, the length of the cord $c= r=25.0 \; m$. By using conversion factor $\frac{\pi \; rad}{180^{\circ}}$ Central angle in radian measure is $\theta = 60.0^{\circ}\times \frac{\pi \; rad}{180^{\circ}}$ Simplify. $\theta = \frac{\pi }{3}\; rad$ (a.) Formula for the length of the arc is $s=r\theta$ Substitute all values. $s=(25.0\;m)(\frac{\pi }{3})$ Simplify. $s=26.2\; m$ correct to one decimal place. (b.) Formula for the area of the sector is $A=\frac{1}{2} r^2 \theta $ Substitute all values $A=\frac{1}{2} (25.0 \; m)^2 (\frac{\pi }{3}) $ Simplify. $A=327 \;m^2$ (Rounded value). (c.) Formula for the area of the segment is $A=\frac{1}{2}r^2\theta-\frac{c\sqrt{4r^2-c^2}}{4}$ From the part (b.) $\frac{1}{2} r^2 \theta = 327\; m^2 $ Substitute all values into the formula. $A=327\;m^2-\frac{(25.0\; m)\sqrt{4(25.0\;m)^2-(25.0\; m)^2}}{4}$ Simplify. $A=327\;m^2-271 \; m^2$ $A=56 \;m^2$. (Rounded value.)
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