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Easy

Question

The surface area of a sphere increases at the rate of 1 sq. ft per second, then the rate of increase of volume when the radius is 3 f t is

  1. 3 cubic ft/sec    
  2. 2 cubic ft/sec    
  3. 2 divided by 3 cubic ft/sec    
  4. 3 divided by 2 cubic ft/sec    

Hint:

The surface area of a sphere is defined as the region covered by its outer surface in three-dimensional space.  A Sphere is a three-dimensional solid having a round shape, just like a circle. The formula of total surface area of a sphere in terms of pi (π) is given by:

Surface area =  π rsquare units

The correct answer is: 3 divided by 2 cubic ft/sec


    We know that,
    Surface area of sphere(S) = 4 πr squared
    fraction numerator d s over denominator d t end fraction space equals space 8 πr dr over dt space space left parenthesis straight d over dx straight x to the power of straight n space equals space nx to the power of straight n minus 1 end exponent right parenthesis
From space the space question space ds over dt space equals 1 sq. ft space per space second
space 8 πr dr over dt space equals space 1
dr over dt space equals fraction numerator 1 over denominator 24 straight pi end fraction space left parenthesis space at space straight r space equals space 3 right parenthesis
Volume space of space sphere left parenthesis straight V right parenthesis space equals space 4 over 3 πr cubed
dv over dt space equals space 4 πr squared dr over dt space left parenthesis straight d over dx straight x to the power of straight n space equals space nx to the power of straight n minus 1 end exponent right parenthesis space equals space 4 straight pi cross times 9 cross times fraction numerator 1 over denominator 24 straight pi end fraction space equals space 36 over 24 space equals space 3 over 2 cubic space ft divided by sec
Thus comma space space the space rate space of space increase space of space volume space when space the space radius space is space 3 space straight f space straight t space is space 3 over 2 cubic space ft divided by sec

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