If a circle has a 3cm radius,
1. By what factor does the circumference change if the radius is tripled?
C = 2pi * radius
Why is the answer correct?
2. By what factor does the area change if the radius is doubled? Explain.
Area = pi * (rad)^2
Why is the answer correct
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1. To find the factor by which the circumference changes when the radius is tripled, we can use the formula for the circumference of a circle: C = 2π * radius.
If the radius is tripled, that means the new radius would be 3 * 3 cm = 9 cm. We can calculate the new circumference using the new radius: C' = 2π * 9 cm = 18π cm.
To find the factor of change, we divide the new circumference by the original circumference: Factor = C' / C = (18π cm) / (2π * 3 cm) = 3.
Therefore, the factor by which the circumference changes when the radius is tripled is 3.
Why is the answer correct?
The answer is correct because we have used the correct formula for the circumference of a circle and have correctly calculated the new circumference by substituting the new radius into the formula. Dividing the new circumference by the original circumference gives us the correct factor of change.
2. To find the factor by which the area changes when the radius is doubled, we can use the formula for the area of a circle: Area = π * (radius)^2.
If the radius is doubled, the new radius would be 2 * 3 cm = 6 cm. We can calculate the new area using the new radius: Area' = π * (6 cm)^2 = 36π cm^2.
To find the factor of change, we divide the new area by the original area: Factor = Area' / Area = (36π cm^2) / (π * (3 cm)^2) = 12.
Therefore, the factor by which the area changes when the radius is doubled is 12.
Why is the answer correct?
The answer is correct because we have used the correct formula for the area of a circle and have correctly calculated the new area by substituting the new radius into the formula. Dividing the new area by the original area gives us the correct factor of change.