Rate of change of area of a circle with respect to radius
(A) When r = 2 cm is 4 \( \pi \) cm2/cm.
(B) When r = 6 cm is 12 \( \pi \) cm2/cm.
(C) When r = 12 cm is 6 \( \pi \) cm2/cm.
(D) When r = 16 cm is 8 \( \pi \) cm2/cm.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (B) only
The area of a circle is \( A=\pi r^2 \), so
\( \displaystyle \frac{dA}{dr}=2\pi r \)
So the rate of change is \( 2\pi r \): \( 4\pi \) at r = 2, \( 12\pi \) at r = 6, \( 24\pi \) at r = 12 and \( 32\pi \) at r = 16 (in cm2/cm).
Checking the statements
(A) When r = 2 cm is 4 \( \pi \) cm2/cm. — Correct. \( 2\pi(2)=4\pi \).
(B) When r = 6 cm is 12 \( \pi \) cm2/cm. — Correct. \( 2\pi(6)=12\pi \).
(C) When r = 12 cm is 6 \( \pi \) cm2/cm. — Incorrect. \( 2\pi(12)=24\pi \), not \( 6\pi \).
(D) When r = 16 cm is 8 \( \pi \) cm2/cm. — Incorrect. \( 2\pi(16)=32\pi \), not \( 8\pi \).
NCERT (2026-27 edition):
Rate of Change of Quantities
