To find the rate of change of the volume of a sphere with respect to its surface area, we first need to express both the volume and the surface area in terms of the radius of the sphere. The volume V of a sphere is given by the formula: V=‌
4
3
Ï€r3 The surface area S of a sphere is given by the formula: S=4Ï€r2
We need to find the rate of change of V with respect to S, which is expressed as ‌
dV
dS
. To do this, we use the chain rule: ‌
dV
dS
=‌
dV
dr
⋅‌
dr
dS
First, we find ‌
dV
dr
: ‌
dV
dr
=‌
d
dr
(‌
4
3
πr3)=4πr2 Next, we find ‌
dS
dr
: ‌
dS
dr
=‌
d
dr
(4πr2)=8πr Now, we need to find ‌
dr
dS
. Since ‌
dS
dr
=8πr, we can write: ‌
dr
dS
=‌
1
8Ï€r
Finally, we substitute ‌
dV
dr
and ‌
dr
dS
back into the chain rule expression: ‌
dV
dS
=(4πr2)⋅(‌
1
8Ï€r
)=‌
r
2
We know from the surface area formula that S=4πr2. Solving for r in terms of S, we get: r2=‌