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Application Of Derivatives

Question
CBSEENMA12034988
Wired Faculty App

The radius of a balloon is increasing at the rate of 10 cm per second. At what rate is the surface area of the balloon increasing when its radius is 15 cm?

Solution

Let r be the radius of the balloon.
therefore space space space space space space dr over dt space equals space 10 space cm divided by sec.                               ...(1)
Let S be the surface area of the balloon.
therefore space space space space space space space space space straight S space equals space 4 space straight pi space straight r squared therefore space space space space space space space space ds over dt space equals space straight d over dt left parenthesis 4 πr squared right parenthesis space equals space 4 space straight pi space straight d over dt left parenthesis straight r squared right parenthesis space equals space 4 space straight pi space open parentheses 2 straight r space dr over dt close parentheses equals space 8 space straight pi space straight r space dr over dt space space space space space space space space space space space space space space space space space space space space space space
                                     equals space 8 πr space cross times space 10 space equals space 80 space straight pi space straight r 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
When r = 15,     dS over dt space equals space 80 space cross times space straight pi space cross times space 15 space equals space 1200 space straight pi
therefore space space space space space space rate space of space increase space of space surface space area space equals space 1200 space straight pi space sq. space cm divided by sec.

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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