Question

the boundary of the shaded region in the given figure consists of three semicircles, the smaller begin equal .if the diameter of the large cm find ​

Answer 1

Diameter of largest semi circle = 28cm

Two small semi circle are equal, hence their diameter will be half of the larger se circle = 28/2 = 14cm

(i) the length of the boundary

Perimeter of semicircle =

$\frac{(\pi)d}{2} + d$

Perimeter of larger semicircle =

$\frac{(\pi) \times 28}{2} + 28 = 14\pi + 28$

Perimeter of smaller semicircle=

$= \frac{(\pi) \times 14}{2} + 14 = 7\pi + 14$

Length of boundary = ( perimeter of larger semicircle - diameter of larger semicircle) + (perimeter of smaller semicircle 1 diameter of smaller semicircle1) +(perimeter of smaller semicircle - diameter of smaller semicircle 2)

$Length of boundary = (14\pi + 28 - 28) + (7\pi + 14 - 14) + (7\pi + 14 - 14) \\ = 14\pi + \pi + 7\pi \\ = 28\pi \\ 28 \times 3.14 \\ = 87.92cm$

(ii) area of shaded region

Area of shaded region = Area of larger semicircle +Area of smaller semicircle1 - Area of smaller semicircle2

Area of semicircle =

$\frac{\pi {r}^{2} }{2}$

$Area \: of \: shaded \: region = \frac{\pi \times {14}^{2} }{2} + \frac{\pi \times {7}^{2} }{2} - \frac{\pi \times {7}^{2} }{2} \\ = \frac{\pi \times196 }{2} = \frac{22 \times 196}{2 \times 7} = 11 \times 28 = 308 {cm}^{2}$