Chapter 11.3, Problem 35PPS

To determine

The area of the shaded region.

Answer to Problem 35PPS

The area of the shaded region is 10.7 $c{m}^2$ .

Explanation of Solution

Given: Two circles enclosed within a rectangle having length 10 cm and breadth 5 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 formula for the area of a rectangle with length l and breadth b is given by-

Area of rectangle $=l\times b$

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

Calculations: The total region in this case is a rectangle having length 10 cm and breadth 5 cm. Area of the rectangle is thus given by-

Total area $=$ Area of rectangle

  $=10\times 5$ $c{m}^2$

  $=50$ $c{m}^2$

The unshaded region consists of two circles which are congruent which is evident from the given figure. Let the diameter of each circle be d. Thus, sum of both diameters is equal to the length of the rectangle. Hence-

  $d+d=10$cm

  $\Rightarrow d=5$cm

Thus, radius of each small circle is given by-

Radius $=\frac{5}{2}=2.5$ $c{m}^2$

Hence, the total area of the unshaded region is equal to twice the area of one circle. Thus-

Unshaded area $=$ $2\times$ (Area of one circle)

  $=2\times \pi {\left(2.5\right)}^2$ $c{m}^2$

  $=2\times 3.14\times 6.25$ $c{m}^2$

  $=39.3$ $c{m}^2$ (approx.)

Hence, total area of the shaded region is given by-

Shaded area $=$ Area of rectangle $-$ Unshaded area

  $=50-39.3$ $c{m}^2$

  $=10.7$ $c{m}^2$

Thus, area of the shaded region is $10.7$ $c{m}^2$