Question

In Fig. there are three semicircles, A, B and C having diameter 3 cm each, and another semicircle E having a circle D with diameter 4.5 cm are shown. Calculate the cost of painting the shaded region at the rate of ₹25 per $c{m}^2$.

Answer 1

Area of the shaded region can be calculated as shown below,

Area of the shaded region = Area of the semi-circle E − area of 2 semi-circles A and C − area of the circle D + area of semi-circle B

Radius of E = 4.5 cm, Radius of A = B = C= 1.5 cm, Radius of D = 2.25 cm.

$Areaofshadedregion=\frac{\pi \times {4.5}^2}{2}-2\times \frac{\pi \times {1.5}^2}{2}-\pi \times {2.25}^2+\frac{\pi \times {1.5}^2}{2}$

$=\frac{\pi \times {4.5}^2}{2}-\frac{\pi \times {1.5}^2}{2}-\pi \times {2.25}^2$

$=\frac{\pi }{2}(20.25-2.25)-\pi \times 5.0625$

$=\frac{\pi }{2}\left(18\right)-\pi \times 5.0625$

$=9\pi -\pi \times 5.0625$

$=\pi (9-5.0625)=3.9375\pi$

Substitute~ $\pi =\frac{22}{7}$, we get,

$Areaofshadedregion=3.9375\times \frac{22}{7}=12.375$

Therefore, area of the shaded region is 12.37 cm²

Now we will find the cost of painting the shaded region at the rate of ₹25per cm².

$Cost=12.375\times 25=309.375=₹309$

Therefore, cost of painting the shaded region is ₹309.