If the market rate per square feet of land is Rs. 1400, how much does the farmer must pay to buy the land from his neighbour for the pond?(Choose the closest option.)
A
Step-by-Step Solution
Key idea: This is an inscribed-circle-in-triangle problem, recognisable because a circle's diameter lies on the base and its arc touches the two equal sides. The trigger is "diameter on base" + "touches equal sides".
Step 1: Analyse the triangle geometry.
Base ft. Altitude from apex to base ft.
Since the triangle is isosceles, the altitude bisects the base. Let be the midpoint of .
ft.
In right : ft.
Step 2: Locate the circle's centre.
The diameter lies on , and the circle touches both equal sides and . By symmetry, the centre must be at the midpoint of .
Step 3: Find the radius .
The circle touches side at point , so and .
Use the area of in two ways:
Equating: ft.
Step 4: Determine which area the farmer buys.
The problem states "half of the circular area will be inside the triangular plot and the other half will be outside". Since the diameter lies on the base, the base splits the full circle into two semi-circles: one above (inside the triangle region) and one below (outside). The farmer buys the outside half.
Step 5: Calculate cost.
Area to buy .
Cost .
Answer: Rs. 3,16,80,000 (Option A).