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Sagot :
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].
[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].
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