The radius of a circle is increasing at the rate of 3 cm/s. The rate at which its circumference is increasing when the radius is 10 cm is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(6\pi\) cm/s
Let r be the radius and C the circumference at time t. Then
\(C = 2\pi r\)
Differentiating with respect to t by the chain rule:
\( \displaystyle \frac{dC}{dt} = 2\pi \frac{dr}{dt} \)
\( = 2\pi \times 3 \)
\( = 6\pi \) cm/s
The rate does not depend on the radius, so it is \(6\pi\) cm/s when r = 10 cm as well.
Why the other options are incorrect
Option (1) \(60\pi\) cm/s — Incorrect. This is \(2\pi r \times 3\) at r = 10, the rate of increase of the area, not the circumference.
Option (2) \(3\pi\) cm/s — Incorrect. This drops the factor 2 in \(C = 2\pi r\).
Option (4) \(20\pi\) cm/s — Incorrect. This is the circumference itself at r = 10 cm, measured in cm, not a rate.