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: Understand the geometry

We have an isosceles triangle with:

• Base = 300 feet
• Height (perpendicular from apex to base) = 200 feet
• Two equal sides

The circle has:

• Diameter entirely on the base
• Half inside the triangle, half outside
• Circumference touches the two equal sides from inside

Step 2: Find the radius of the circle

Since the circle's diameter is on the base and half is inside/half outside, the center of the circle is on the base, and the circle is tangent to the two equal sides.

For an isosceles triangle with base $b = 300$ and height $h = 200$:

• The apex is at coordinates $(150, 200)$ if we place the base from $(0, 0)$ to $(300, 0)$
• The two equal sides make equal angles with the base

The equation of one of the equal sides (say, left side from $(0, 0)$ to $(150, 200)$): The slope is $\frac{200}{150} = \frac{4}{3}$

The line equation: $4x - 3y = 0$

For a circle centered at $(150, 0)$ with radius $r$ to be tangent to this line: Distance from center to line = $r$

$$\frac{|4(150) - 3(0)|}{\sqrt{16 + 9}} = \frac{600}{5} = 120 = r$$

So the radius is 120 feet.

Step 3: Calculate the area to be purchased

Half the circular area is outside the triangle: $$\text{Area outside} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (120)^2 = \frac{1}{2} \pi \times 14400 = 7200\pi \text{ sq ft}$$

Step 4: Calculate the cost

$$\text{Cost} = 7200\pi \times 1400$$ $$= 10,080,000\pi$$ $$= 10,080,000 \times 3.14159$$ $$\approx 31,668,067$$ $$\approx \text{Rs. } 3,16,68,000$$

This is closest to Rs. 3,16,80,000.

Answer: (1) Rs. 3,16,80,000 

Answer 2

Setup: Isosceles triangle with base = 300 ft, height = 200 ft. Circle's diameter lies on the base, with half the area inside and half outside the triangle.

Find the radius: Place the base from $(0,0)$ to $(300,0)$ with apex at $(150, 200)$.

The left side from $(0,0)$ to $(150,200)$ has equation: $4x - 3y = 0$

The circle is centered at $(150, 0)$ (midpoint of base) and tangent to both equal sides.

Distance from center $(150, 0)$ to line $4x - 3y = 0$: $$r = \frac{|4(150) - 3(0)|}{\sqrt{16+9}} = \frac{600}{5} = 120 \text{ feet}$$

Calculate area outside triangle: $$\text{Area} = \frac{1}{2}\pi r^2 = \frac{1}{2}\pi(120)^2 = 7200\pi \text{ sq ft}$$

Calculate cost: $$\text{Cost} = 7200\pi \times 1400 = 10,080,000\pi \approx 31,668,067$$

Closest to Rs. 3,16,80,000.

Answer: (1) Rs. 3,16,80,000