Use differentials to estimate

Question

The Radius of a spear is measured to be 5.5 inches. If the measurement is correct two with the 0.03 inch, use differentials to estimate the propagated error in the volume of the sphere. V=4/3pi r^3 find the percent error

Answer 1

We're using differentials to determine the approximate error.

So, we are looking for dV = f(radius)

If we use the definition for differentials that dy = f ' (x) dx, then

dV = V ' (r) dr

dV = 4 pi r^2 dr

Since r = 5.5 and the error is 0.03 inches

dV = 4 pi (5.5)^2 * 0.03 inches^3

If you want to turn this into an error percentage, you could calculate dV / V * 100.