XAT 2015 Quant & DI Q6 of 33: Mensuration

What is to be found?

We have to find the lowest cost to construct 50% of the total road.

Data Given

A circular road is constructed outside a square field. So, the road is in the shape of a circular ring.

If we have to determine the lowest cost of constructing the road, we have to select the smallest circle that can be constructed outside the square.

Therefore, the inner circle of the ring should circumscribe the square.

Perimeter of the square = 200 ft.

Therefore, side of the square field = 50 ft.

Compute the inner and outer radii of the circular ring

The diagonal of the square field is the diameter of the circle that circumscribes it.
Measure of the diagonal of the square of side 50 ft = 50\\sqrt2\\) ft.
Therefore, inner diameter of the circular road = 50\\sqrt2\\).
Hence, inner radius of the circular road = 25\\sqrt2\\) ft.
Then, outer radius = 25\\sqrt2\\) + 7\\sqrt2\\) = 32\\sqrt2\\)

Compute the area of the circular ring

The area of the circular road
= π r_o² - π r_i², where r_o is the outer radius and r_i is the inner radius.

= \\frac{22}{7}\\) × {(32 \\sqrt2\\))² – (25\\sqrt2\\))²}
= \\frac{22}{7}\\)×2×(32 + 25)×(32 – 25)
= 2508 sq. ft.

If per sq. ft. cost is Rs. 100, then cost of constructing the road = 2508 × 100 = Rs.2,50,800.

Cost of constructing 50% of the road = 50% of the total cost = \\frac{250800}{2}\\) = Rs.1,25,400

  The correct answer is Choice B.