Question

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

A
Sπ
B
12Sπ
C
14Sπ
D
4Sπ

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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