Final answer:
To find the rate at which the volume changes with respect to the radius, differentiate the volume formula with respect to the radius. Substitute the radius value into the derivative equation to find the rate of change. Multiply the rate of change by the change in radius to find how much the volume increases.
Step-by-step explanation:
To find the rate at which the volume changes with respect to the radius, we need to differentiate the volume formula with respect to the radius. The volume of a sphere is given by V = (4/3)πr^3, where V is the volume and r is the radius. Differentiating this formula with respect to r gives us dV/dr = 4πr^2.
a. To find the rate at which the volume changes with respect to the radius when r = 11 in, we substitute r = 11 into the derivative equation: dV/dr = 4π(11)^2 = 484π. Therefore, the rate at which the volume changes with respect to the radius when r = 11 in is 484π cubic inches per unit change in radius.
b. To find how much the volume increases when the radius changes from 11 to 11.1, we need to multiply the rate from part a by the change in radius: ΔV = (484π)(11.1-11) = 484π(0.1) = 48.4π cubic inches.