The side length of a cube was measured to be 16.4 mm with a possible error of 0.13 mm .
Use differentials to estimate the maximum error in the calculated volume. Do not round your answer.

To estimate the maximum error in the calculated volume using differentials, we start with the volume formula for a cube: , where is the side length. The differential gives us the change in volume with respect to a change in side length :

Given and :

Thus, the estimated maximum error in the calculated volume is approximately .

The maximum error calculated indicates how sensitive the volume of a cube is to small changes in side length. With the side length of the cube only slightly varying, we see significant implications in the final volume value. It’s fascinating how such tiny measurements can yield substantial differences in volume, which is crucial in applications like manufacturing or laboratory settings where precision matters immensely!