The diameter of a circle is increasing at the rate of 2 cm/sec. When its radius is 4 cm, the rate of change of increase of its area is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( 8\pi\ cm^2/sec \)
The diameter is 2r, so the radius increases at half the rate of the diameter.
\( \displaystyle \frac{d(2r)}{dt}=2 \)
\( \displaystyle \frac{dr}{dt}=1\ \text{cm/sec} \)
\( \displaystyle A=\pi r^2 \)
\( \displaystyle \frac{dA}{dt}=2\pi r\,\frac{dr}{dt} \)
At r = 4:
\( \displaystyle \frac{dA}{dt}=2\pi(4)(1)=8\pi\ \text{cm}^2\text{/sec} \)
Tip: When a rate is given for the diameter, halve it before using the radius formula; using 2 cm/sec as dr/dt would give 16π.