A square garden measures 30 m by 30 m. Two flower beds in the shape of a quarter circles are made with two opposite vertices as centres. If the radius of each of the quarter circles is 7 m, find the portion of the garden with grass.

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Calculate the area of the square garden.
The area of a square is given by the formula:
\[ \text{Area} = \text{side} \times \text{side} \]
Given that the side of the square garden is 30 m:
\[ \text{Area} = 30 \, \text{m} \times 30 \, \text{m} = 900 \, \text{m}^2 \]

### Step 2: Calculate the area of one quarter circle.
The area of a full circle is given by the formula:
\[ \text{Area} = \pi r^2 \]
Since we need the area of a quarter circle, we will divide the area of the full circle by 4:
\[ \text{Area of quarter circle} = \frac{\pi r^2}{4} \]
Given that the radius \( r \) is 7 m:
\[ \text{Area of quarter circle} = \frac{\pi (7 \, \text{m})^2}{4} = \frac{\pi \times 49 \, \text{m}^2}{4} \]

### Step 3: Substitute the value of \(\pi\).
Using \(\pi \approx \frac{22}{7}\):
\[ \text{Area of quarter circle} = \frac{\frac{22}{7} \times 49}{4} = \frac{22 \times 49}{28} = \frac{1078}{28} = 38.5 \, \text{m}^2 \]

### Step 4: Calculate the area of two quarter circles.
Since there are two quarter circles:
\[ \text{Area of two quarter circles} = 2 \times \text{Area of one quarter circle} = 2 \times 38.5 \, \text{m}^2 = 77 \, \text{m}^2 \]

### Step 5: Calculate the area of the garden with grass.
To find the area of the garden that remains with grass, subtract the area of the flower beds from the area of the square garden:
\[ \text{Area with grass} = \text{Area of square garden} - \text{Area of flower beds} \]
\[ \text{Area with grass} = 900 \, \text{m}^2 - 77 \, \text{m}^2 = 823 \, \text{m}^2 \]

### Final Answer:
The portion of the garden with grass is \( 823 \, \text{m}^2 \).

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