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: Calculate the area of the triangular plot. The area of a triangle is:

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

Substitute the base = 300 feet and height = 200 feet:

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

Step 2: Calculate the radius of the circular pond. Since the circumference of the circle touches the two equal sides of the triangle, the radius \(r\) is the inradius of the triangle. The inradius is given by:

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

Calculate the semi-perimeter \(s\). Let the two equal sides of the triangle each be \(x\). Using the Pythagoras 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:

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

The inradius is:

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

Step 3: Calculate the area of the circular pond. The area of a circle is:

Area \(= \pi r^2\)

Substitute \(r = 75\):

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

Since half the 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: Calculate the cost. The cost of the land outside is:

Cost = Outside Area × Rate per square feet

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

Answer: Rs. 4,25,60,000