Match List-I with List-II, where each rate of change is with respect to the radius r, at r = 3 cm.
| List-I | List-II | ||
| (A) | Rate of change of the area of a circle | (I) | \(6\pi\) |
| (B) | Rate of change of the circumference of a circle | (II) | \(36\pi\) |
| (C) | Rate of change of the volume of a sphere | (III) | \(2\pi\) |
| (D) | Rate of change of the surface area of a sphere | (IV) | \(24\pi\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A)-(I), (B)-(III), (C)-(II), (D)-(IV)
Each rate is the derivative with respect to r, evaluated at r = 3.
(A) Rate of change of the area of a circle
This matches (I) \(6\pi\).
\( \displaystyle \frac{d}{dr}(\pi r^2) = 2\pi r = 6\pi \)
(B) Rate of change of the circumference of a circle
This matches (III) \(2\pi\).
\( \displaystyle \frac{d}{dr}(2\pi r) = 2\pi \)
(C) Rate of change of the volume of a sphere
This matches (II) \(36\pi\).
\( \displaystyle \frac{d}{dr}\left(\frac{4}{3}\pi r^3\right) = 4\pi r^2 = 36\pi \)
(D) Rate of change of the surface area of a sphere
This matches (IV) \(24\pi\).
\( \displaystyle \frac{d}{dr}(4\pi r^2) = 8\pi r = 24\pi \)