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Question

The slant height of a cone is fixed as  7 cm. If the rate of increase in its height is 0.3 cm/sec. Then the rate of increase in volume when the height is 4cm

  1. fraction numerator pi over denominator 2 end fraction c c divided by s e c    
  2. pi c c divided by s e c    
  3. fraction numerator pi over denominator 5 end fraction c c divided by s e c    
  4. fraction numerator 33 pi over denominator 10 end fraction c c divided by s e c    

hintHint:

Volume of cone = 1 third πr squared straight h

The correct answer is: fraction numerator 33 pi over denominator 10 end fraction c c divided by s e c


     Given : Slant height (l) = 7cm
    Rate of increase in its height is 0.3 cm/sec. 
    Height = 4cm

    Using Pythagoras Theorem
    H y p o t e n u s e squared space equals space B a s e squared space plus space P e r p e n d i c u l a r squared
    7 squared space equals space r squared space plus space 4 squared
    r squared space equals space 7 squared space minus space 4 squared
r squared space equals space 49 space minus space 16
r squared space equals space 33
    We know that, 
    V o l u m e space o f space c o n e space equals space 1 third πr squared straight h
Rate space of space increase space in space volume space of space cone space equals space 1 third πr squared space cross times space left parenthesis rate space of space straight h right parenthesis
equals space 1 third straight pi space cross times 33 space cross times 0.3 space cm divided by straight s
equals space straight pi cross times 3.3 cm cubed divided by straight s
equals space fraction numerator 33 straight pi over denominator 10 end fraction cm cubed divided by straight s
    Thus, the rate of increase in volume of cone is fraction numerator 33 straight pi over denominator 10 end fraction c c divided by s e c 

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