Let's analyze this problem using calculus concepts. First, we know the formula for the volume of a sphere, given by:
V=‌πr3To find how the volume change rate
‌ affects the radius
r, we need to differentiate the volume formula with respect to time
t. Using the chain rule, we have:
‌=‌(‌πr3)=4πr2‌Here,
‌ represents the rate of change of the radius
r. Given that
‌ is constant, we can rearrange the formula to solve for
‌ :
‌=‌ This equation shows that
‌ is inversely proportional to
r2, which comprises the surface area of the sphere. The surface area
S of a sphere is given by:
S=4Ï€r2Substitute the surface area in the formula for
‌ :
‌=‌This shows that the rate at which the radius increases,
‌, is inversely proportional to the surface area
S of the sphere. Constant rate of increase in volume leads to an inverse proportionality between the rate of radius increase and the sphere's surface area.