Precalculus (10th Edition)

Published by Pearson
ISBN 10: 0-32197-907-9
ISBN 13: 978-0-32197-907-0

Chapter 1 - Graphs - 1.4 Circles - 1.4 Assess Your Understanding - Page 39: 50


$(36\pi - 72) \text{ square units}$

Work Step by Step

Using the standard form $x^2+y^2=r^2$ (where $r$ is the radius) as basis, then $r^2=36 \implies r=6$. The area $A$ of a circle is give by the formula $A=\pi{r^2}$ where $r=\text{radius}$ Hence, the area of the circle is: $$A=\pi(6^2)=36\pi \text{ square units}$$ The area of the shaded region is equal to the difference between the area of the circle and the area of the square. Let the side of the square be $a$. The diagonal divides the square into two congruent right triangles, with the base and height equal to $a$. The diagonal of the square is the diameter of the circle. Since the radius is $6$ units, then the diameter is $12$ units. Hence, the diagonal of the square is $12$ units long. The area of a square is equal to the square of its side. Using Pythagorean Theorem givesx_{y}: \begin{align*} \text{(base)}^2 + \text{(height)}² &= \text{(hypotenuse)}^2\\ a^2 + a^2 &= 12^2\\ 2a^2 &= 144\\ a^2 &= 72\\ \end{align*} Thus, the area of the square is $72 \text{ square units}$. This means that the area of the shaded region is: \begin{align*} \text{Area of the shaded region} &= \text{area of the circle} - \text{area of the square}\\ &= (36\pi - 72) \text{ square units} \end{align*}
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