Question

There is a circular park near Ram’s house. In this park, there are rectangular and circular paths of width 1.5 m. The rectangular paths pass through center of the park. The local community decided to put grass beds on the 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 laying grass beds on the entire pathway at 0.1 paise per square cm.(Consider overlapping areas as squares)

Answer 1

For circular path, the inner radius = 8 m

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

Area of the circular path = $\pi$ $\left[\right(outerradius{)}^2-\left(innerradius{)}^2\right]$

$=\pi ({9.5}^2-8^2)=\left(3.14\right)\times 17.5m\times 1.5m=82.425m^2$ (Using identity $a^2-b^2=(a+b)(a-b))$ (1 mark)

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 = (82.425 + 72) $m^2$ = 154.425 $m^2$ (1 mark)

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

So, overlapping area = 5 $\times$ $(1.5{)}^2$ = 5 $\times$ $\left(2.25\right)$ = 11.25 $m^2$

Hence, Total area = (154.425 – 11.25) $m^2$ = 143.175 $m^2$ (1 mark)

Cost of laying 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 = ₹10

Total cost = (₹ 10 / $m^2$) $\times$ 143.175 $m^2$ = ₹1431.75 (1 mark)