5000 beach balls are to be constructed to have a diameter of 1 ft with a thickness of 0.03 inches.  Use differentials to estimate how much material will be needed for this project.  Then compare with the true value.

Solution

The formula for the volume of 5000 beach balls is

We want to estimate the additional volume that is needed to go from a diameter of 1 ft to a diameter of 0.03 inches.  In terms of radius, half the diameter, we want r to go from 0.5 to 0.5 + 0.015.

We have 

We have r  =  0.5 and Dr  =  0.015.  We approximate DV with dV to get

To find DV exactly, assuming that we mean that the inside of the balloon is 1 ft in diameter with an additional 0.03 inches of material, we find

Notice how closely the the approximation is to the actual value.

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