Chapter 12, Problem 26CR

To determine

To calculate: The area of shaded region which contains two concentric circles.

Answer to Problem 26CR

The area of shaded region is $20\pi$ .

Explanation of Solution

Given information:

The coordinates of point Aare$\left(-6,8\right)$ .

The coordinates of point B are $\left(10,0\right)$ .

Formula used:

The distance between two points by using the distance formula, which is an application of the Pythagoras theorem.

D$=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

  $(x_1,y_1)$ arecoordinates of first point in the line.

  $(x_2,y_2)$ arecoordinates of second point in the line.

Area of a circle: $A=\pi r^2$

Where “r” is radius of circle.

Calculation:

  

It is given that,

The coordinates of point Aare$\left(-6,8\right)$ .

The coordinates of point B are $\left(10,0\right)$ .

The coordinates of midpoints of $\overline{AB}$ are $\left(\left(\frac{-6+10}{2}\right),\left(\frac{8+0}{2}\right)\right)=\left(2,4\right)$ .

If O is the center of circle then OM is radius.

  $\begin{array}{l}OM=\sqrt{{\left(2-0\right)}^2+{\left(4-0\right)}^2}\\ OM=\sqrt{{\left(2\right)}^2+{\left(4\right)}^2}\\ OM=\sqrt{4+16}\\ OM=r=\sqrt{20}\end{array}$

Shaded Area:

  $\begin{array}{l}A=\pi r^2\\ A=\pi {\left(\sqrt{20}\right)}^2\\ A=20\pi \end{array}$