Chapter 1.4, Problem 48AYU

To determine

To find: The area of the shaded region

Answer to Problem 48AYU

The area of the shaded region is $41.04$ Square units

Explanation of Solution

Given:

  

Calculation:

The standard form of an equation of a circle with center at the origin $(0,0)$ and radius $r$

  $x^2+y^2=r$ .

On comparing the given equation with the standard form, the value of $r$ as $6$

The area of a circle is $\pi r^2$ .Substitute $6$ for $r$ in the formula

  $Area=\pi {\left(6\right)}^2$

  $=36\pi$

  $\approx 113.04square$ $units$

Since the radius of the circle is 6,the diameter will be $2\left(6\right)$ or $12$ .

The diameter of the circle is equivalent to the diagonal of the square

In order to find the side of square, use the formula $d=a\sqrt{2}$ .Substitute $12$ for $d$ in the formula

  $12=a\sqrt{2}$

Divide each side of the equation by $\sqrt{2}$

  $\frac{12}{\sqrt{2}}=\frac{a\sqrt{2}}{\sqrt{2}}$

  $\frac{12}{\sqrt{2}}=a$

The area of a square is given by the formula

  $Area=a^2$ , where $a$ is the side length of the square

Substitute $\frac{12}{\sqrt{2}}$ for $a$ in the formula

  $Area={\left(\frac{12}{\sqrt{2}}\right)}^2$

  $=72$ Square units

The area of the shaded region is the difference between the area of the circle and the area of the square

The area of the shaded region $=113.04-72$

  $\approx 41.04$

Conclusion:

Hence, the area of the shaded region is $41.04$ Square units