If the area of a circle is doubled, by what factor does its circumference change?

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Multiple Choice

If the area of a circle is doubled, by what factor does its circumference change?

Explanation:
Doubling the area means the new area is twice the original. Since area equals πr^2, the new radius r' satisfies πr'^2 = 2πr^2, so r' = sqrt(2) · r. The circumference depends linearly on radius: C = 2πr. With the new radius, the circumference becomes 2πr' = 2π(sqrt(2)r) = sqrt(2) · (2πr). So the circumference increases by a factor of sqrt(2).

Doubling the area means the new area is twice the original. Since area equals πr^2, the new radius r' satisfies πr'^2 = 2πr^2, so r' = sqrt(2) · r. The circumference depends linearly on radius: C = 2πr. With the new radius, the circumference becomes 2πr' = 2π(sqrt(2)r) = sqrt(2) · (2πr). So the circumference increases by a factor of sqrt(2).

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