Question

Near Ted’s house, there is a circular park. In this park, there are rectangular and circular paths with width 1.5 m. The rectangular paths pass through center of the park. The local community decided to put grass beds on entire pathway. The inner radius of the circular path is 8 m. If the distance between any of the gates and the circular path is 2.5 m, find the cost of putting grass beds on the entire pathway at 0.1 paise per square cm.

• A. Rs. 1185
• B. Rs. 1545
• C. Rs. 1432.5
• D. Rs. 1522.5

Answer 1

The correct option is C

Rs. 1432.5

For circular path, the inner radius = 8 m

So, the outer radius = (8 + 1.5) m = 9.5 m

Area of the circular path = $\pi$ $\left[\right(\text{outer radius}{)}^2-\left(\text{inner radius}{)}^2\right]$

$=\pi ({9.5}^2-8^2)=\left(22/7\right)\times 17.5m\times 1.5m=82.5m^2$

(Using identity $a^2-b^2=(a+b)(a-b))$

Radius of the park = Outer radius of circular path + Distance between the path and any gate

= (9.5 + 2.5) m = 12 m

Length of the rectangular pathways = Diameter of the park = 24 m

So, area of 2 rectangular pathways = 2 $\times$ (lb) = 2 $\times$ 24 m $\times$ 1.5 m = 72 $m^2$

Total area = (82.5 + 72) $m^2$ = 154.5 $m^2$

But we have taken 5 areas twice (shaded in the figure). These are the areas where rectangular paths are overlapping each other or the circular path. These overlappings are nearly square in shape.

So, overlapped area = 5 $\times$ $(1.5m{)}^2$ = 11.25 $m^2$

Total area hence = (154.5 – 11.25) $m^2$ = 143.25 $m^2$

Cost of putting grass beds = 0.1 paisa per $c{m}^2$ ​

So, for 1 $m^2$ i.e. 10000 $c{m}^2$, cost = (0.1 $\times$ 10000) paise = 1000 paise = Rs. 10

Total cost = (Rs. 10 / $m^2$) $\times$ 143.25 $m^2$ = Rs. 1432.5