MHT CET · Maths · Application of Derivatives

If the surface area of a spherical balloon of radius \(6 \mathrm{~cm}\) is increasing at the rate \(2 \mathrm{~cm}^2 / \mathrm{sec}\), then the rate of increase in its volume in \(\mathrm{cm}^3 / \mathrm{sec}\) is

A \(16\)
B \(6\)
C \(12\)
D \(8\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(6\)

Step-by-step Solution

Detailed explanation

Surface area, \(S=4 \pi r^2\)
\(\begin{aligned}
& \therefore \quad \frac{\mathrm{dS}}{\mathrm{dt}}=8 \pi \mathrm{r} \frac{\mathrm{dr}}{\mathrm{dt}} \\
& \Rightarrow 2=8 \pi \mathrm{r} \frac{\mathrm{dr}}{\mathrm{dt}} \\
& \Rightarrow \frac{\mathrm{dr}}{\mathrm{dt}}=\frac{1}{4 \pi \mathrm{r}}...(i) \\
& \text { Volume, } V=\frac{4}{3} \pi r^3 \\
& \therefore \quad \frac{\mathrm{dV}}{\mathrm{dt}}=\frac{4}{3} \times 3 \pi \mathrm{r}^2 \times \frac{\mathrm{dr}}{\mathrm{dt}} \\
& =4 \pi r^2 \times \frac{1}{4 \pi r}... [From (i)] \\
& =\mathrm{r} \\
& =6 \mathrm{~cm}^3 / \mathrm{sec} \\
&
\end{aligned}\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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