Question

A path of 4 m width runs round a semi-circular grassy plot whose circumference is $163\frac{3}{7}m$. Find:

(i) The area of the path
(ii) The cost of gravelling the path at the rate of Rs. 1.50 per square metre
(iii) The cost of turfing the plot at the rate of 45 paise per $m^2$.

Answer 1

Here radius of semi circular grassy plot = r

And

Radius of semi circular grassy plot with 4 m wide path = R = ( r + 4 ) m

Given : Circumference of semi circular grassy plot = 11447 m

We know : Circumference of semicircle = πr + 2 r , So

⇒ r π + 2 = 11447⇒ r 227 + 2 = 11447⇒ r 22 + 147 = 11447⇒ r 367 = 11447⇒ r = 11447 × 736⇒ r =31.78
So,

Radius of semi circular grassy plot with 4 m wide path = R = ( r + 4 ) = 31.78 + 4 = 35.78 m

1 ) Area of path = Area of semi circular grassy plot with 4 m wide path - Area of semi circular grassy plot

We know area of semicircle = π r22

Area of path =227 ×35.7822 - 227 ×31.7822 = 11 ×1280.20847 - 11 ×1009.96847 = 2011.756 - 1587.0932 = 424.6628≈424.66 m2 ( Ans )
2 ) Given : Rate of gravelling the path at Rs1.50 per metre square.

So,

Cost of gravelling the path at the = 424.66 ×1.50 = Rs .636.99 ≈ Rs. 637 ( Ans )

3 ) Rate of trufing the plot at the rate of 45 paise per m.

So,

Cost of trufing the plot = 1587.0932 × 0.45 = Rs .714.19194 ≈ Rs. 714.20