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Application Of Derivatives

Question
CBSEENMA12035009
Wired Faculty App

The volume of a cube is increasing at the rate of 8 cm3/s. How fast is the surface area increasing when the length of an edge is 12 cm?

Solution
Let V be volume of cube of side x.
therefore space space space space space space space space space space space space space straight V space equals space straight x cubed
From given condition,
                dV over dt space equals space 8 space cm cubed divided by straight s
therefore space space space space straight d over dt left parenthesis straight x cubed right parenthesis space equals space 8 space space space space space space space space space space space space rightwards double arrow space space space space 3 straight x squared dx over dt space equals space 8 space space space rightwards double arrow space space space space dx over dt space equals fraction numerator 8 over denominator 3 straight x squared end fraction space space space space space space space space space space space space space... left parenthesis 1 right parenthesis
Let S be surface area of cube
therefore space space space space space straight S space equals space 6 straight x squared
Rate of increase of surface area  = dS over dt space equals space straight d over dt left parenthesis 6 straight x squared right parenthesis
                                             equals space 12 straight x space dx over dt space equals space 12 straight x space cross times space fraction numerator 8 over denominator 3 straight x squared end fraction space space space space space space open square brackets because space space of space left parenthesis 1 right parenthesis close square brackets equals space 32 over straight x When space straight x space equals space 12 comma space space rate space of space increase space of space surface space area space space equals space 32 over 12 equals space 8 over 3 cm squared divided by straight s.

Tips: -

Since the rate of change is represented by derivative with respect to time, it is taken to be positive if the quantity is increasing and negative if the quantity is decreasing.

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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