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Maths-

General

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

3 cubic ft/sec
2 cubic ft/sec
$2 / 3$ cubic ft/sec
$3 / 2$ cubic ft/sec

hint 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 = 4 π r2 square 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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