The shape of a garden is rectangular in the middle and semi circular at the ends as shown in the diagram. Find the area and the perimeter of this garden [Length of rectangle is `20-(3.5+3.5)` metres].

Options

A. ` 129.5, 47`
B. `129.5, 48`
C. `127.7, 49`
D. `189.5, 50`

Explanation

To solve the problem of finding the area and perimeter of the garden, which is rectangular in the middle and semi-circular at the ends, we can follow these steps:

### Step 1: Understand the Dimensions
The total length of the garden is given as 20 meters. The ends of the garden are semi-circles with a combined diameter of 7 meters (3.5 meters on each side). Therefore, the length of the rectangle can be calculated as:
\[ \text{Length of Rectangle} = 20 - (3.5 + 3.5) = 20 - 7 = 13 \text{ meters} \]

### Step 2: Calculate the Area of the Rectangle
The area of a rectangle is given by the formula:
\[ \text{Area of Rectangle} = \text{Length} \times \text{Breadth} \]
Here, the length is 13 meters and the breadth (which is the diameter of the semi-circles) is 7 meters. Thus:
\[ \text{Area of Rectangle} = 13 \times 7 = 91 \text{ square meters} \]

### Step 3: Calculate the Area of the Semi-Circles
The area of a semi-circle is given by:
\[ \text{Area of Semi-Circle} = \frac{1}{2} \pi r^2 \]
The radius \( r \) of the semi-circle is half of the diameter:
\[ r = \frac{7}{2} = 3.5 \text{ meters} \]
Now substituting the value of \( r \):
\[ \text{Area of Semi-Circle} = \frac{1}{2} \times \frac{22}{7} \times (3.5)^2 \]
Calculating \( (3.5)^2 \):
\[ (3.5)^2 = 12.25 \]
Now substituting back:
\[ \text{Area of Semi-Circle} = \frac{1}{2} \times \frac{22}{7} \times 12.25 \]
Calculating this gives:
\[ \text{Area of Semi-Circle} = \frac{22 \times 12.25}{14} = \frac{271.5}{14} = 19.25 \text{ square meters} \]
Since there are two semi-circles, the total area of the semi-circles is:
\[ \text{Total Area of Semi-Circles} = 2 \times 19.25 = 38.5 \text{ square meters} \]

### Step 4: Calculate the Total Area of the Garden
Now, adding the area of the rectangle and the total area of the semi-circles:
\[ \text{Total Area} = \text{Area of Rectangle} + \text{Total Area of Semi-Circles} \]
\[ \text{Total Area} = 91 + 38.5 = 129.5 \text{ square meters} \]

### Step 5: Calculate the Perimeter of the Garden
The perimeter of the garden consists of the two lengths of the rectangle and the circumference of the two semi-circles:
\[ \text{Perimeter} = 2 \times \text{Length of Rectangle} + \text{Circumference of Semi-Circles} \]
The circumference of a full circle is given by:
\[ \text{Circumference} = 2 \pi r \]
Thus, for a semi-circle:
\[ \text{Circumference of Semi-Circle} = \pi r \]
Substituting the radius:
\[ \text{Circumference of Semi-Circles} = \frac{22}{7} \times 3.5 = 22 \text{ meters} \]
Now substituting back into the perimeter formula:
\[ \text{Perimeter} = 2 \times 13 + 22 = 26 + 22 = 48 \text{ meters} \]

### Final Results
- **Total Area of the Garden:** 129.5 square meters
- **Total Perimeter of the Garden:** 48 meters