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 114: 65


$1900\text{ square yards}$

Work Step by Step

The area of the lot is equal to the sum of the areas of the triangle and the sector. First, find the length of the radius of the circle/sector. Note that the radius is actualy the hypotenuse of the right triangle. Hence, ithe radius can be computed using the Pythagorean Theroem to obtain: Let $x$ = radius/hypotenuse Then. \begin{align*} x^2&=30^2+40^2\\ x^2&=900+1600\\ x^2&=2500\\ x&=\sqrt{2500}\\ x&=50\text{ yards} \end{align*} RECALL: (1) The area $A$ of a triangle is given by the formula $A=\frac{1}{2}bh$ where $b=$ base and $h=$ height. (2) The area $A$ of a sector is given by the formula $A=\frac{\theta}{360^o}\pi{r^2}$ where $\theta=$central angle measure in degrees and $r$=radius of the circle. Thus, we have: \begin{align*} \text{Area of the lot} &= \text{Area of the triangle} + \text{Area of the sector}\\ &=\frac{1}{2}(30)(40) + \frac{60^0}{360^o}\pi(50^2)\\\\ &=15(40) + \frac{1}{6}2500\pi\\\\ &=600+1,308.9969389957\\\\ &\approx1900 \text{ square yards} \end{align*}
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