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Short Answer Type

21. 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.

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22.
The volume of a spherical balloon is increasing at the rate of 25 cm³/sec. Find the rate of change of its surface area at the instant when its radius is 5 cm.

Let V be volume of sphere of radius r.

therefore space space space space space space space space space space space space straight V space equals space 4 over 3 πr cubed

From given condition,

dV over dt space equals space 25 space cm cubed divided by straight s comma space space space space space space space space space space space space space space space space space space therefore space space space space straight d over dt open parentheses 4 over 3 πr cubed close parentheses space equals space 25 space space space space space rightwards double arrow space space space space 4 over 3 straight pi. space 3 straight r. space dr over dt space equals space 25

therefore space space space space space space space dr over dt space equals space fraction numerator 25 over denominator 4 πr squared 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 space space... left parenthesis 1 right parenthesis

Let S be surface area of balloon.

therefore space space space space space space space space space straight S space equals space 4 πr squared

Rate of increase of surface area =

dS over dt space equals space straight d over dt left parenthesis 4 πr squared right parenthesis

equals space 8 space πr space dr over dt space equals space 8 space πr space cross times space fraction numerator 25 over denominator 4 πr squared end fraction 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 space left parenthesis 1 right parenthesis close square brackets

equals space 50 over straight r

When r =5, rate of increase of surface area =

50 over 5 space equals space 10 space cm squared divided by straight s.

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Long Answer Type

23. The radius of a spherical soap bubble is increasing at the rate of 0.3 cms^–1. Find the rate of change of its (i) volume (ii) surface area when the radius is 8 cm.

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24. The radius of a spherical soap bubble is increasing at the rate of 0.2 cms^–1. Find the rate of change of its (i) volume (ii) surface area, when the radius is 4 cm.

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25. The radius of a spherical soap bubble is increasing at the rate of 0.4 cms^–1 Find the rate of change of its (i) volume (ii) surface area, when the radius is 10 cm.

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Short Answer Type

26.

A ballon which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate of which the radius of the balloon is increasing when the radius is 15 cm.

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27.

The volume of a cube is increasing at a rate of 9 cubic centimeters per second. How fast is the surface area increasing when the length of an edge is 10 centimeters?

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28.

The volume of a cube is increasing at a rate of 7 cubic centimeters per second. How fast is the surface area increasing when the length of an edge is 12 centimeters?

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29. 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?

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30.

The length x of a rectangle is decreasing at the rate of 2 cm/s and the width y is increasing at the rate of 2 cm/s. When x = 12 cm and y = 5 cm, find the rate of change of (a) the perimeter and (b) the area of the rectangle.

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