A rectangular park is 100 m by 50 m. It is surrounded by semicircular flower beds all round . Find the cost of levelling the semicircular flower beds at 60 paise per `m^(2)`.

Options

A. Rs. 31425
B. Rs. 28260
C. Rs. 352. 40
D. None of these

Answer

To solve the problem, we need to find the area of the semicircular flower beds surrounding the rectangular park and then calculate the cost of leveling them. Here’s a step-by-step solution:

### Step 1: Understand the Dimensions
The rectangular park has dimensions:
- Length = 100 m
- Width = 50 m

### Step 2: Determine the Radius of the Semicircles
The semicircular flower beds are located on the longer sides (100 m) and the shorter sides (50 m) of the rectangle.

- The radius of the semicircles on the longer sides (length) will be half of the width of the rectangle:
\[ \text{Radius} = \frac{50}{2} = 25 \text{ m} \]

- The radius of the semicircles on the shorter sides (width) will be half of the length of the rectangle:
\[ \text{Radius} = \frac{100}{2} = 50 \text{ m} \]

### Step 3: Calculate the Area of the Semicircles
1. **Area of the two semicircles on the longer sides:**
\[ \text{Area of one semicircle} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (25)^2 = \frac{1}{2} \pi (625) = 312.5\pi \text{ m}^2 \]
Therefore, the area for two semicircles:
\[ \text{Total Area} = 2 \times 312.5\pi = 625\pi \text{ m}^2 \]

2. **Area of the two semicircles on the shorter sides:**
\[ \text{Area of one semicircle} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (50)^2 = \frac{1}{2} \pi (2500) = 1250\pi \text{ m}^2 \]
Therefore, the area for two semicircles:
\[ \text{Total Area} = 2 \times 1250\pi = 2500\pi \text{ m}^2 \]

### Step 4: Calculate the Total Area of the Flower Beds
Now, we add the areas of the semicircles:
\[ \text{Total Area of Flower Beds} = 625\pi + 2500\pi = 3125\pi \text{ m}^2 \]

### Step 5: Calculate the Cost of Leveling
The cost of leveling is given as 60 paise per square meter. First, we need to convert the area into numerical form using \(\pi \approx 3.14\):
\[ \text{Total Area} \approx 3125 \times 3.14 \approx 9825 \text{ m}^2 \]

Now, calculate the cost:
\[ \text{Cost} = 9825 \times 0.60 = 5895 \text{ paise} \]
Convert paise to rupees (100 paise = 1 rupee):
\[ \text{Cost in Rupees} = \frac{5895}{100} = 58.95 \text{ rupees} \]

### Final Answer
The cost of leveling the semicircular flower beds is **₹58.95**.

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