Chapter 11.3, Problem 34PPS

To determine

The area of the shaded region.

Answer to Problem 34PPS

The area of the shaded region is 77 $m{m}^2$ .

Explanation of Solution

Given: Two circles enclosed within a larger circle of diameter 14 mm 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: The total region in this case is the larger circle whose diameter is 14 mm.

Hence, radius of the larger circle will be equal to-

Radius = $\frac{14}{2}=7$mm.

Thus, area of the total region is given by-

Total area $=\pi {\left(7\right)}^2$ $m{m}^2$

  $=\frac{22}{7}\times {\left(7\right)}^2$ $m{m}^2$

  $=154$ $m{m}^2$

The unshaded region consists of two circles enclosed within the larger circle. From the figure it is clear that the both the unshaded circles are congruent and the sum of the diameters of both the circles is equal to the diameter of the larger circle. Let the diameter of both the unshaded circles be d.

Thus,

  $d+d=14$

  $\Rightarrow d=7$

Hence, the diameter of each small circle is 7 $m{m}^2$ . Hence, the radius of each small circle is-

Radius = $\frac{7}{2}$mm.

Therefore total area of the unshaded region is given by-

Total unshaded area $=$ $2\times$ (Area of one small circle)

  $=2\times \pi {\left(\frac{7}{2}\right)}^2$ $m{m}^2$

  $=2\times \frac{22}{7}\times {\left(\frac{7}{2}\right)}^2$ $m{m}^2$

  $=11\times 7$ $m{m}^2$

  $=77$ $m{m}^2$

Thus, the area of the shaded region is given by-

Shaded area $=$ Area of large circle $-$ Unshaded Area

  $=154-77$ $m{m}^2$

  $=77$ $m{m}^2$

Thus, total shaded area is $77$ $m{m}^2$ .