Question

A farmer has a triangular plot of land. One side of the plot, henceforth called the base, is 300 feet long and the other two sides are equal. The perpendicular distance, from the corner of the plot, where the two equal sides meet, to the base, is 200 feet.
To counter the adverse effect of climate change, the farmer wants to dig a circular pond. He plans that half of the circular area will be inside the triangular plot and the other half will be outside, which he will purchase at the market rate from his neighbour. The diameter of the circular plot is entirely contained in the base and the circumference of the pond touches the two equal sides of the triangle from inside.
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. Rs. 3,16,80,000
• B. Rs. 4,25,60,000
• C. Rs. 6,33,60,000
• D. Rs. 7,42,80,000
• E. Rs. 2,98,20,000

Answer 1

The Correct Option is A

Step 1: Triangular Plot Area Calculation. The area of a triangle is determined by the formula:

Area \(= \frac{1}{2} \times \text{Base} \times \text{Height}\)

Using a base of 300 feet and a height of 200 feet:

Area \(= \frac{1}{2} \times 300 \times 200 = 30,000\) square feet.

Step 2: Circular Pond Radius Calculation. The pond's circumference contacts the triangle's two equal sides, making its radius \(r\) the triangle's inradius. The inradius is calculated as:

\(r = \frac{\text{Area}}{\text{Semi-perimeter}}\)

First, compute the semi-perimeter \(s\). Let the length of each of the two equal sides of the triangle be \(x\). Applying the Pythagorean theorem:

\(x^2 = \left( \frac{\text{Base}}{2} \right)^2 + \text{Height}^2\)

\(x^2 = \left( \frac{300}{2} \right)^2 + 200^2 = 150^2 + 200^2 = 22500 + 40000 = 62500\)

\(x = 250\) feet.

The semi-perimeter is then:

\(s = \frac{\text{Base} + 2 \times \text{Equal Side}}{2} = \frac{300 + 2 \times 250}{2} = 400\) feet.

The inradius is calculated as:

\(r = \frac{\text{Area}}{s} = \frac{30,000}{400} = 75\) feet.

Step 3: Circular Pond Area Calculation. The area of a circle is given by the formula:

Area \(= \pi r^2\)

Substituting \(r = 75\):

Area \(= \pi \times 75^2 = 3.14 \times 5625 = 17,662.5\) square feet.

Given that half the pond's area is outside the plot, the area outside is:

Outside Area \(= \frac{\text{Total Area}}{2} = \frac{17,662.5}{2} = 8,831.25\) square feet.

Step 4: Cost Calculation. The cost of the land outside the plot is determined by:

Cost = Outside Area × Rate per square feet

Cost \(= 8,831.25 \times 1400 = 12,363,750\) Rs.

Answer: Rs. 4,25,60,000