Three horses are grazing within a semi-circular field. In the diagram given below, AB is the diameter of the semi-circular field with centre at O. Horses are tied up at P, R and S such that PO and RO are the radii of semi-circles with centres at P and R respectively, and S is the centre of the circle touching the two semi-circles with diameters AO and OB. The horses tied at P and R can graze within the respective semi-circles and the horse tied at S can graze within the circle centred at S. The percentage of the area of the semi-circle with diameter AB that cannot be grazed by the horses is nearest to
Answer: B
Consider the diagram below.
Let r be the radius of the smaller semi-circles and s be the radius of the smaller circle.
OS = 2r − s
PS = r + s
PO = r
But, ∆PSO is a right-angled triangle.
∴ PS2 = PO2 + SO2
(r + s)2 = r2 + (2r − s)2
∴ r² + s² + 2rs = r² + 4r² + s² − 4rs
∴ Total area not grazed = π(2r)2 -
= 2πr2 - πr2 - πr2
= πr2
∴ Required percentage = × 100
= × 100
≈ 28%
Hence, option (b).