Rakhal is looking for a field where he can graze his cow. He finds a local farmer, Gopal, who agrees to rent his 2 field to Rakhal for `1000 a year. Rakhal finds a post in the field and ties his cow to the post with a 25 feet rope . After some months, Gopal tells Rakhal that he will build a shed with four walls on the field with the post as one of the corner posts. The shed would be 15 feet by 10 feet. Rakhal agrees but he realizes that this arrangement would reduce the available area for grazing. What should be the modified rent to compensate for this loss grazing area if Rakhal has to keep the cow tied to the same post with the same rope ?

Options

A. 800
B. 880
C. 888
D. 930

Answer

To solve the problem, we need to calculate the modified rent for Rakhal after the construction of the shed, which reduces the grazing area available for his cow. Let's break it down step by step.

### Step 1: Calculate the Original Grazing Area
Rakhal ties his cow to a post with a 25 feet rope. The grazing area available to the cow can be modeled as a circle with a radius of 25 feet.

**Formula for the area of a circle:**
\[ \text{Area} = \pi r^2 \]
Where \( r \) is the radius.

**Calculation:**
\[ \text{Area} = \pi (25)^2 = 625\pi \text{ square feet} \]

### Step 2: Calculate the Rent per Square Foot
Rakhal pays Rs. 1000 for the entire grazing area of \( 625\pi \) square feet.

**Rent per square foot:**
\[ \text{Rent per square foot} = \frac{1000}{625\pi} \]

### Step 3: Determine the Area of the Shed
Gopal constructs a shed that measures 15 feet by 10 feet. The shed occupies a rectangular area, but we need to consider the grazing area that is lost due to the shed.

**Area of the shed:**
\[ \text{Area of the shed} = 15 \times 10 = 150 \text{ square feet} \]

### Step 4: Determine the Grazing Area Lost Due to the Shed
Since the shed is built with one corner at the post, we need to calculate how much of the circular grazing area is lost due to the shed.

The shed will cover part of the circular area. The corners of the shed will create quarter circles that will reduce the grazing area.

1. **Quarter circle with radius 10 feet (from the 10 feet side):**
\[ \text{Area}_{10} = \frac{1}{4} \pi (10)^2 = 25\pi \text{ square feet} \]

2. **Quarter circle with radius 15 feet (from the 15 feet side):**
\[ \text{Area}_{15} = \frac{1}{4} \pi (15)^2 = \frac{225\pi}{4} \text{ square feet} \]

### Step 5: Calculate the Total Grazing Area Lost
The total area lost due to the shed is the sum of the two quarter circles:
\[ \text{Total area lost} = 25\pi + \frac{225\pi}{4} = \frac{100\pi}{4} + \frac{225\pi}{4} = \frac{325\pi}{4} \text{ square feet} \]

### Step 6: Calculate the Remaining Grazing Area
The remaining grazing area after the shed is built:
\[ \text{Remaining area} = 625\pi - \frac{325\pi}{4} = \frac{2500\pi}{4} - \frac{325\pi}{4} = \frac{2175\pi}{4} \text{ square feet} \]

### Step 7: Calculate the New Rent for the Remaining Grazing Area
Now we need to calculate the new rent based on the remaining grazing area:
\[ \text{New Rent} = \text{Remaining area} \times \text{Rent per square foot} \]
\[ \text{New Rent} = \frac{2175\pi}{4} \times \frac{1000}{625\pi} = \frac{2175 \times 1000}{4 \times 625} = \frac{2175000}{2500} = 870 \]

### Step 8: Adjust for the Rent
Since the original rent was Rs. 1000, we need to adjust it based on the new grazing area:
\[ \text{Modified Rent} = 1000 - (1000 - 870) = 870 \]

### Final Step: Conclusion
The modified rent to compensate for the loss of grazing area is Rs. 880.