CUET · MATHS · PYQ PAPER 2025
The rate of change of volume of a sphere with respect to its surface area, when the radius is 6 cm is:
- A \(1 \mathrm{~cm}^3 / \mathrm{cm}^2\)
- B \(6 \mathrm{~cm}^3 / \mathrm{cm}^2\)
- C \(2 \mathrm{~cm}^3 / \mathrm{cm}^2\)
- D \(3 \mathrm{~cm}^3 / \mathrm{cm}^2\)
Answer & Solution
Correct Answer
(D) \(3 \mathrm{~cm}^3 / \mathrm{cm}^2\)
Step-by-step Solution
Detailed explanation
\(V = \frac{4}{3} \pi r^3\), \(\frac{dV}{dr} = 4 \pi r^2\) \(A = 4 \pi r^2\), \(\frac{dA}{dr} = 8 \pi r\) \(\frac{dV}{dA} = \frac{dV/dr}{dA/dr} = \frac{4 \pi r^2}{8 \pi r} = \frac{r}{2}\) For \(r = 6\): \(\frac{dV}{dA} = \frac{6}{2} = 3 \, \mathrm{cm}^3 / \mathrm{cm}^2\)
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