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.)
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.)
- Rs. 3,16,80,000
- Rs. 4,25,60,000
- Rs. 6,33,60,000
- Rs. 7,42,80,000
- Rs. 2,98,20,000
The Correct Option is A
Solution and Explanation
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
Top Questions on Geometry
- Find the equation of the circle touching the x-axis at \( (9, 0) \) and the line \( y = 14 \).
- Given the line equation \( Ax + By + C = 0 \), which passes through the point \( (-10, 7) \) and is perpendicular to the line \( 11x - 8y - 16 = 0 \), find \( C \).
The center of a circle $ C $ is at the center of the ellipse $ E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $, where $ a>b $. Let $ C $ pass through the foci $ F_1 $ and $ F_2 $ of $ E $ such that the circle $ C $ and the ellipse $ E $ intersect at four points. Let $ P $ be one of these four points. If the area of the triangle $ PF_1F_2 $ is 30 and the length of the major axis of $ E $ is 17, then the distance between the foci of $ E $ is:
- Let for two distinct values of $ p $, the lines $ y = x + p $ touch the ellipse $ E: \frac{x^2}{4} + \frac{y^2}{9} = 1 $ at the points $ A $ and $ B $. Let the line $ y = x $ intersect $ E $ at the points $ C $ and $ D $. Then the area of the quadrilateral $ ABCD $ is equal to:
- The axis of a parabola is the line $ y = x $ and its vertex and focus are in the first quadrant at distances $ \sqrt{2} $ and $ 2\sqrt{2} $ units from the origin, respectively. If the point $ (1, k) $ lies on the parabola, then a possible value of $ k $ is:
Questions Asked in XAT exam
- In an 8-week course, the teacher conducts a test every week, and the scores are in the range of 1–4. There are only two students enrolled in the course, \(R\) and \(S\). The following conditions are given:
- \(R\) and \(S\) scored the same on the first test.
- From the second test onwards, \(R\) consistently scored the same (a non-zero score).
- The total of the first three test scores of \(R\) equals the total of the first two test scores of \(S\).
- From the fifth test onwards, \(S\) scored the same as \(R\).
- The scores of \(S\) from the first two tests and all the remaining tests follow a geometric progression.
- XAT - 2025
- Algebra
- Arya, a graduate from a reputable institute, got a job in an IT company but became bored after a year. Her best friend S, who worked at the same company, joined a top-tier B-school for an MBA, which tempted Arya to pursue the same path. Her friend assured her that an MBA would boost her career prospects and salary. Arya started preparing for an MBA while continuing her job, but the challenge of balancing preparation with work made her consider leaving the job.
- XAT - 2025
- Analytical Decision Making
- There’s a community 30 km outside of the main city. Mr. S started a grocery business in the community after winning the bid by offering way above the rent initially drafted by the community council. The community council, led by Mr. D, agreed to the offer with a performance review every three years. Mr. S also agreed to provide an additional 15% of his grocery sales to the council, banking on the prospect of selling in higher volumes. After setting up his business, Mr. S noticed that SUV owners in the community usually bought goods in bulk from the city once a week and used his store primarily for daily necessities and occasional big purchases like mixer grinders.
- XAT - 2025
- Analytical Decision Making
- Four reviewers (R1, R2, R3, R4) are in charge of reviewing products from companies A, B, C, and D. Each reviewer provides a rating between 1 and 5 for the products. Due to a technical glitch, the original ratings were deleted, and only the averages calculated for each product (A, B, C, D) and each reviewer (R1, R2, R3, R4) were saved. The data available is as follows:Tasks:
A B C D Average 3 4 4 ? 4 3 ? 5 ? 4 ? 3 3 ? 4 ? ? ? ? 4.25 4 4 4 4.25
1. What are the missing values in the table?
2. How were the missing values calculated?
3.Verify the correctness of the given averages for both rows and columns. - A bought a phone from a store and paid \(\frac{1}{6}\) of the price using UPI, \(\frac{1}{3}\) of the price in cash, and the remaining balance a year later. He also paid 10% interest on the remaining balance after one year. What was the original price of the phone?
- XAT - 2025
- Percentage