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Question

The rate of change of volume of sphere with respect to its surface area $$S$$ is 


A
Sπ
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B
12Sπ
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C
14Sπ
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D
4Sπ
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Solution

The correct option is C $$\cfrac { 1 }{ 4 } \sqrt { \cfrac { S }{ \pi } } $$
Volume of sphere with radius $$r$$ is $$\cfrac { 4 }{ 3 } \pi { r }^{ 3 }$$ and surface area $$S=\pi { r }^{ 2 }$$
$$\therefore \cfrac { dV }{ dr } =\cfrac { 4 }{ 3 } \pi \left( 3{ r }^{ 2 } \right) =4\pi { r }^{ 2 }$$
Now $$S=4\pi { r }^{ 2 }$$
$$\cfrac { dS }{ dr } =4\pi (2r)$$
$$\therefore \cfrac { dV }{ dS } =\cfrac { \cfrac { dV }{ dS }  }{ \cfrac { dS }{ dr }  } =\cfrac { 4\pi { r }^{ 2 } }{ 8\pi r } =\cfrac { r }{ 2 } $$
$$=\cfrac { 1 }{ 2 } (r)...(1)$$
Now $$S=4\pi { r }^{ 2 }$$
$$\therefore { r }^{ 2 }=\cfrac { S }{ 4\pi  } $$
$$\quad r=\cfrac { 1 }{ 2 } \left( \sqrt { \cfrac { S }{ \pi  }  }  \right) \quad $$
$$\therefore$$ Rate of change of volume w,r,to
$$S=\cfrac { 1 }{ 2 } \left[ \cfrac { 1 }{ 2 } \left( \sqrt { \cfrac { S }{ \pi  }  }  \right)  \right] =\cfrac { 1 }{ 4 } \left( \sqrt { \cfrac { S }{ \pi  }  }  \right) $$

Mathematics

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