Chapter 11.3, Problem 36PPS

To determine

The area of the shaded region.

Answer to Problem 36PPS

The area of the shaded region is 7.9 $m^2$ .

Explanation of Solution

Given: Two shaded sectors in a circle of radius 3 m as shown below-

  

Concept Used: The formula for the area of the sector of a circle with central angle $\theta$ and radius r is given by-

Area of sector $=\frac{\theta }{360°}\times \pi r^2$

Total angle at the centre of any circle is $360°$ .

Calculations: The unshaded region consists of two sectors of the circle with central angle $130°$ . Thus, total central angle included in the unshaded region

  $=2\times 130°$

  $=260°$

Hence, total central angle included within the shaded region is

  $=360°-260°$

  $=100°$

The shaded region consists of two congruent sectors each with radius 3 m. Let the central angle of both the shaded sectors be x. Thus,

  $x+x=100°$

  $\Rightarrow x=50°$

Hence, the central angle of each shaded sector is $50°$ . Thus, total area of the shaded region is given by-

Shaded area $=$ $2\times$ (Area of one shaded sector)

  $=2\times \frac{50°}{360°}\times \pi {\left(3\right)}^2$ $m^2$

  $=\frac{100°}{360°}\times 3.14\times 9$ $m^2$

  $=\frac{5}{18}\times 3.14\times 9$ $m^2$

  $=7.9$ $m^2$ (approx.)

Thus, area of the shaded region is $7.9$ $m^2$ .