(a) Sharmila has drawn a circle with a radius of 5 cm. Similarly, Prakash has drawn a circle with a radius of 7 cm. Whose circle has a greater area and by how much? (Use [tex]\(\pi = 3.14\)[/tex])



Answer :

To determine whose circle has the greater area and by how much, we will follow a systematic approach by using the formula for the area of a circle:

[tex]\[ \text{Area} = \pi \times r^2 \][/tex]

where [tex]\( r \)[/tex] is the radius of the circle.

### Step-by-Step Solution:

Step 1: Calculate the area of Sharmila's circle

Given:
- Radius of Sharmila's circle, [tex]\( r_1 = 5 \)[/tex] cm
- Value of [tex]\( \pi = 3.14 \)[/tex]

Using the area formula:
[tex]\[ \text{Area}_{\text{Sharmila}} = \pi \times (r_1)^2 \][/tex]
[tex]\[ \text{Area}_{\text{Sharmila}} = 3.14 \times (5)^2 \][/tex]
[tex]\[ \text{Area}_{\text{Sharmila}} = 3.14 \times 25 \][/tex]
[tex]\[ \text{Area}_{\text{Sharmila}} = 78.5 \, \text{cm}^2 \][/tex]

Step 2: Calculate the area of Prakash's circle

Given:
- Radius of Prakash's circle, [tex]\( r_2 = 7 \)[/tex] cm
- Value of [tex]\( \pi = 3.14 \)[/tex]

Using the area formula:
[tex]\[ \text{Area}_{\text{Prakash}} = \pi \times (r_2)^2 \][/tex]
[tex]\[ \text{Area}_{\text{Prakash}} = 3.14 \times (7)^2 \][/tex]
[tex]\[ \text{Area}_{\text{Prakash}} = 3.14 \times 49 \][/tex]
[tex]\[ \text{Area}_{\text{Prakash}} = 153.86 \, \text{cm}^2 \][/tex]

Step 3: Determine whose area is greater

To find whose circle has the greater area, we compare the results:

[tex]\[ \text{Area}_{\text{Sharmila}} = 78.5 \, \text{cm}^2 \][/tex]
[tex]\[ \text{Area}_{\text{Prakash}} = 153.86 \, \text{cm}^2 \][/tex]

Clearly, [tex]\( 153.86 \, \text{cm}^2 \)[/tex] is greater than [tex]\( 78.5 \, \text{cm}^2 \)[/tex]. Therefore, Prakash's circle has the greater area.

Step 4: Calculate the difference in area

To find the difference in area between Prakash's and Sharmila's circles:

[tex]\[ \text{Difference in area} = \text{Area}_{\text{Prakash}} - \text{Area}_{\text{Sharmila}} \][/tex]
[tex]\[ \text{Difference in area} = 153.86 \, \text{cm}^2 - 78.5 \, \text{cm}^2 \][/tex]
[tex]\[ \text{Difference in area} = 75.36 \, \text{cm}^2 \][/tex]

### Conclusion:

- Prakash's circle has a greater area.
- The difference in area between Prakash's circle and Sharmila's circle is [tex]\( 75.36 \, \text{cm}^2 \)[/tex].