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Question

a)The volume of a growing spherical cell is V=4/3pie(r)^3, where the radius is measured in micrometers . Find the average rates of change of V with respect to r when r changes from 8 to 12micrometers , from 8 to 10micrometers, and from 8 to 8.3

b)Find the instantaneous rate of change of V with respect to r when r =8micrometers

Answer 1

First, find the derivative of V with respect to r:
dV/dr = (4/3)pi d/dr[r^3] = (4/3)pi (3r^2) = 4 pi r^2
a) Multiply both sides by dr:
dV = 4 pi (r^2) dr = 4pi (8^2) [10-8] = 4pi(64)(2) (microm)^3
dV = 4pi (r^2) dr = 4pi(8^2) [8.3-8] = 4pi(64)(0.3) (microm)^3

b) dV/dr|r=8 = 4pi(8^2) = 4(64)pi