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=‌πr3The 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
‌. To do this, we use the chain rule:
‌=‌⋅‌First, we find
‌ :
‌=‌(‌πr3)=4πr2Next, we find
‌ :
‌=‌(4πr2)=8πr Now, we need to find
‌. Since
‌=8πr, we can write:
‌=‌Finally, we substitute
‌ and
‌ back into the chain rule expression:
‌=(4πr2)⋅(‌)=‌We know from the surface area formula that
S=4Ï€r2. Solving for
r in terms of
S, we get:
r2=‌‌‌⇒‌‌r=√‌ Substituting this back into
‌, we get:
‌=‌√‌=‌⋅‌√‌=‌√‌Therefore, the correct answer is:
Option D:
‌√‌