Chapter 11.3, Problem 40PPS

To determine

The area of the shaded region.

Answer to Problem 40PPS

The area of the shaded region is 47.7 $c{m}^2$ .

Explanation of Solution

Given: Two concentric circles where the larger circle has diameter of 9 cm as shown below-

  

Concept Used: The formula for the area of circle with radius r is given by-

Area of circle $=\pi r^2$

The area of any shaded region can be found by subtracting the total unshaded region from the total region.

Calculations: Let the radius of the inner circle and the length of the circular track be x. From the given figure, the diameter of the inner circle and two times the length of the circular track adds up to the 9 cm. Thus,

  $x+2x+x=9$

  $\Rightarrow 4x=9$

  $\Rightarrow x=\frac{9}{4}$

Thus, radius of the inner circle is $\frac{9}{4}$cm.

The diameter of the larger circle is 9 cm. Thus, radius of the larger circle will be-

Radius $=\frac{9}{2}$cm.

Hence, area of the larger circle

  $=\pi \times {\left(\frac{9}{2}\right)}^2$ $c{m}^2$

  $=\frac{81}{4}\pi$ $c{m}^2$

Area of the smaller circle

  $=\pi \times {\left(\frac{9}{4}\right)}^2$ $c{m}^2$

  $=\frac{81}{16}\pi$ $c{m}^2$

Thus, area of the circular track

  $=\frac{81}{4}\pi -\frac{81}{16}\pi$ $c{m}^2$

  $=\frac{243}{16}\pi$ $c{m}^2$

  $=\frac{243}{16}\times 3.14$ $c{m}^2$

  $=47.7$ $c{m}^2$ (approx.)

Thus, the area of the shaded region is 47.7 $c{m}^2$ .