## Trigonometry (11th Edition) Clone

Published by Pearson

# Chapter 3 - Radian Measure and the Unit Circle - Section 3.2 Applications of Radian Measure - 3.2 Exercises - Page 113: 63c

#### Answer

The area of the portion of the circle bounded by the arc and the chord is $~~619.7~ft^2$

#### Work Step by Step

We can find the radius $r$ of the curve: $sin~21^{\circ} = \frac{50~ft}{r}$ $r = \frac{50~ft}{sin~21^{\circ}}$ $r = 139.5~ft$ We can convert the angle to units of radians: $42^{\circ}\cdot~\frac{\pi~rad}{180^{\circ}} = 0.733~rad$ We can find the total area $A_t$ of the sector: $A_t = \frac{r^2~\theta}{2},~~~$ where $\theta$ is measured in radians $A_t = \frac{(139.5~ft)^2~(0.733~rad)}{2}$ $A_t = 7132.2~ft^2$ We can find the length $L$ of the line from the center of the circle which bisects the chord: $\frac{50~ft}{L} = tan~21^{\circ}$ $L = \frac{50~ft}{tan~21^{\circ}}$ $L = 130.25~ft$ We can find the area $A_2$ of half of the triangular section: $A_2 = \frac{1}{2}(50~ft)(130.25~ft) = 3256.25~ft^2$ The total area of the triangular section is double this area. The total area of the triangular section is $(2)(3256.25~ft^2) = 6512.5~ft^2$ We can find the area of the portion of the circle bounded by the arc and the chord: $Area = 7132.2~ft^2 - 6512.5~ft^2 = 619.7~ft^2$ The area of the portion of the circle bounded by the arc and the chord is $~~619.7~ft^2$

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