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Application Of Derivatives

Question
CBSEENMA12034976
Wired Faculty App

A ballon which always remain spherical, has a variable diameter 3 over 2 left parenthesis 2 straight x plus 3 right parenthesis. Determine the rate of charge of its volume with respect to x.

Solution

Let r be the radius of the balloon.
 From given condition,         diameter =3 over 2 left parenthesis 2 straight x plus 3 right parenthesis
therefore space space space space space space 2 straight r space equals space 3 over 2 left parenthesis 2 straight x plus 3 right parenthesis space space space space space space space space space space space space rightwards double arrow space space space space straight r space equals space 3 over 4 left parenthesis 2 straight x plus 3 right parenthesis
Let V be the volume of spherical balloon.
therefore space space space space space straight V space equals space fraction numerator 4 straight pi over denominator 3 end fraction straight r cubed space equals space fraction numerator 4 straight pi over denominator 3 end fraction. space space space 27 over 64 left parenthesis 2 straight x plus 3 right parenthesis cubed space space space space space space space space space space space space space space rightwards double arrow space space space space straight V space equals space fraction numerator 9 straight pi over denominator 16 end fraction left parenthesis 2 straight x plus 3 right parenthesis cubed therefore space space space space rate space of space change space of space volume space straight w. straight r. straight t. space straight x space equals space dV over dx 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 space space space space space space space space space space space space space space space space space space space space space space space space equals space straight d over dx open square brackets fraction numerator 9 straight pi over denominator 16 end fraction left parenthesis 2 straight x plus 3 right parenthesis cubed close square brackets space equals space fraction numerator 9 straight pi over denominator 16 end fraction. straight d over dx left parenthesis 2 straight x plus 3 right parenthesis cubed 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 space space space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator 9 straight pi over denominator 16 end fraction.3 space space left parenthesis 2 straight x plus 3 right parenthesis squared space. space 2 space equals space fraction numerator 27 space straight pi over denominator 8 end fraction left parenthesis 2 straight x plus 3 right parenthesis squared.

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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