SOLUTION: The volume of a sphere is given by V(r)=4/3pie(r^3)
a) Find the average rate of change of volume with respect to radius as the radius changes from 10cm to 15 cm.
b) Find the rat
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Question 181345: The volume of a sphere is given by V(r)=4/3pie(r^3)
a) Find the average rate of change of volume with respect to radius as the radius changes from 10cm to 15 cm.
b) Find the rate of change of volume when the radius is 8 cm
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
a) Find the average rate of change of volume with respect to radius as the radius changes from 10cm to 15 cm.
To find the average rate of change between two values of the independent variable, evaluate the function at each of the values, giving you two ordered pairs that are on the graph of the function. Then, the formula for calculating the slope of a line, 
, will give you the average rate of change. Specifically, for your problem:
= \frac{4}{3} \pi r^3 )
You need to calculate:
 - V(10)}{15 - 10}= \frac {\frac{4}{3} \pi (15)^3 - \frac{4}{3} \pi (10)^3}{5})
b) Find the rate of change of volume when the radius is 8 cm
Now you are looking for the instantaneous rate of change at a point. The instantaneous rate of change is the value of the first derivitive evaluated for the given value of the independent variable.
=\frac{4}{3} \pi r^3)
So:
=4 \pi r^2)
Hence, you need to evaluate
=4 \pi 8^2)
The arithmetic is up to you. I would leave my answers in terms of rational expressions including 
rather than trying to present a numerical approximation.
John
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