(a) The volume V of a growing spherical cell is V = 4/3 pi r3

Question

(a) The volume V of a growing spherical cell is V = 4/3 pi r^, where the radius is measured in micrometers (1 µm =
10^-6m). Find the average rate of change of V with respect to r when r changes from 5 to each of the following. (Round your answers to one decimal place.)

(i) 5 to 8 µm
(ii) 5 to 6 µm
(iii) 5 to 5.1 µm


(b) Find the instantaneous rate of change of V with respect to r when r = 5 µm. (Round your answer to one decimal place.)

Answer 1

Average rate of change can be described as the ratio of change to the time over which the change took place.  For a volume, this would be [V(b) - V(a)] / (b-a), where a and b are times and V(a) and V(b) are functions describing the volume at t=a and t=b. 

 

However, instead of time, the problem asks for the rate of change with respect to the radius.

 

V = 4/3*pi*r^3

 

Let's solve part (i), which you can use as a template for the remaining parts.

The average rate of change of V is

 

[V(8)-V(5)] / (8-5).

=[4/3*pi*8^3 - 4/3*pi*5^3] / 3

= 4/3*pi*(512-125)/3

=4/3*pi*387/3

=516*pi

 

The instantaneous rate of change can be found by differentiating the formula for volume:

 

dV/dr = 4*pi*r^2

=4*pi*5^2

 

the answer will be in (μm^3)