AP EAMCET · Maths · Application of Derivatives

The volume of a spherical balloon is increasing at the rate of $30 cc$ per minute. Find the rate of change of surface area of the balloon, when its radius is $6 cm$ .

A $5 {cm}^{2} \cdot \min^{- 1}$
B $30 {cm}^{2} \cdot \min^{- 1}$
C $10 {cm}^{2} \cdot \min^{- 1}$
D $20 {cm}^{2} \cdot \min^{- 1}$

Verified Solution

Answer & Solution

Correct Answer

(C) $10 {cm}^{2} \cdot \min^{- 1}$

Step-by-step Solution

Detailed explanation

⇒v=volume of sphere of radius r ⇒v=43π r3 ⇒dvdt=30 cc/min ⇒ddt43π r3=30 ⇒43π 3 r2drdt=30 ⇒drdt=304π r2 1…

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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The volume of a spherical balloon is increasing at the rate of 30cc per minute. Find the rate of change of surface area of the balloon, when its radius is 6 cm.

Step-by-step Solution

The volume of a spherical balloon is increasing at the rate of $30 cc$ per minute. Find the rate of change of surface area of the balloon, when its radius is $6 cm$ .