Question

Volume \(\quad\) The radius \(r\) and height \(h\) of a right circular cylinder are measured with possible errors of \(4 \%\) and \(2 \%,\) respectively. Approximate the maximum possible percent error in measuring the volume.

Short answer

The maximum possible percent error in measuring the volume is approximately \(10\% \).

Step-by-step solution

- The volume formula

Write down the formula for the volume of a cylinder \(V = \pi r^{2}h\).

- Determine the relative error of the volume

Since we're looking for the percent error in the volume, we need to find how the volume changes with respective to changes in radius and height. This can be done using the formula for relative error, \(dV/V = 2 (dr/r) + (dh/h)\), which is obtained by differentiating the volume formula with respect to each variable and then factoring.

- Substitute given percent errors and solve

Substitute the given percent errors for \(dr/r = 4\% = 0.04\) and \(dh/h = 2\% = 0.02\) into the formula for relative error that we derived. We get \(dV/V = 2(0.04) + 0.02 = 0.10\).

- Convert to percent

The relative error for the volume obtained in step 3 is in decimal. Therefore, in order to get the answer in percent form, we multiply by 100, hence our answer is \(0.10 × 100 = 10 \%\).