Question

The cost of planting the grass in a circular park @ Rs. 5 per m2 is Rs. 27720. A path of uniform width runs around the park. The cost of gravelling the path @ Rs. 3.50 per m2. Is Rs. 10780. Find the cost of fencing the path on both sides @ Rs. 3 per m.

Answer 1

There is a circular park, and a path running around it. We have two circles. Inner circle denotes park and outer circle is the boundary of the path.
Let the radius of inner circle be 'r' and radius of outer circle be 'R'

Cost of planting grass in the park:

Rs 5 for 1 m² area
Rs 1 for 1/5 m² area
Rs 27720 for 27720 X 1/5 = 5544 m² area
Area of the park is 5544 m²
π r² = 5544        [ Area of a circle = π r²]
22/7 x r² = 5544
r² = 5544 x 7/22
r² = 1764
r = √1764 = 42m

Cost of gravelling the path
Rs 3.50 for 1 m²
Rs 1 for 1/3.50 m²
Rs 10780 for 10780 X 1/3.50 = 3080 m²
Area of path = 3080 m²
Area of Path = Area of outer boundary minus area of inner boundary
Area of path = π R² - π r²
                     = π (R² - r²)
3080 = π (R² - r²)
3080 = 22/7 (R² - 1764)
3080 x 7/22 = R² - 1764
980 = R²- 1764
R² = 2744
R = √2744 = 52.38 m
cost of fencing = (circumference of outer circle - circumference of inner circle) x rate per m.
(2πR - 2πr) x 3
2π (52.38 - 42) X 3
= 2 X 22/7 X 10.38 X 3 = Rs 195.74