Question

The surface area of a spherical bubble is increasing at the rate of 2 cm²/sec. Find the rate at which the volume of the bubble is increasing at the instant its radius is 6 cm.

Solution

Let S be the surface of the spherical bubble of radius r.
$therefore space space space space space space space straight S space equals space 4 πr cubed$
From given condition,
$dS over dt space equals space 2 space cm squared divided by sec comma space space space space space space space space therefore space space space space straight d over dt left parenthesis 4 πr squared right parenthesis space equals space 2$
$rightwards double arrow space space space space space space 8 space straight pi space straight r space dr over dt space equals space 2 space space space space space space rightwards double arrow space space space space dr over dt space equals space fraction numerator 1 over denominator 4 πr end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis$
Let V be the volume of the spherical bubble.
$therefore space space space space space space straight V space equals space space 4 over 3 πr cubed$
Rate of increase of volume = $dV over dt space equals space straight d over dt open parentheses fraction numerator 4 πr cubed over denominator 3 end fraction close parentheses$
$equals space 4 over 3 straight pi. space 3 straight r squared space dr over dt space equals space 4 πr squared space dr over dt space equals space 4 πr squared space cross times space space fraction numerator 1 over denominator 4 πr end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space space of space left parenthesis 1 right parenthesis close square brackets space space space equals space straight r space when space straight r space equals space 6 comma space space rate space of space increase space of space volume space space equals space 6 space cm cubed divided by sec.$

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Application Of Derivatives

The surface area of a spherical bubble is increasing at the rate of 2 cm²/sec. Find the rate at which the volume of the bubble is increasing at the instant its radius is 6 cm.

Some More Questions From Application of Derivatives Chapter

The radius of a circle is increasing uniformly at the rate of 4 cm per second Find the rate at which the area of the circle is increasing when the radius is 8 cm.

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For Daily Free Study Material Join wiredfaculty

Application Of Derivatives

The surface area of a spherical bubble is increasing at the rate of 2 cm2/sec. Find the rate at which the volume of the bubble is increasing at the instant its radius is 6 cm.

Some More Questions From Application of Derivatives Chapter

The radius of a circle is increasing uniformly at the rate of 4 cm per second Find the rate at which the area of the circle is increasing when the radius is 8 cm.

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

The surface area of a spherical bubble is increasing at the rate of 2 cm²/sec. Find the rate at which the volume of the bubble is increasing at the instant its radius is 6 cm.