The cubit is an ancient unit of length based on the distance between the elbow and the tip of the middle finger of the measurer. Assume that the distance ranged from 43 to 53 cm, and suppose that ancient drawings indicate that a cylindrical pillar was to have a length of 9 cubits and a diameter of 2 cubits. For the stated range, what are the lower value and the upper value, respectively, for (a) the cylinder’s length in meters, (b) the cylinder’s volume in cubic meters? (c) What is the percent uncertainty in the volume?

For operations with uncertainty, we’ll follow **two different rules** depending on whether we're **adding/subtracting** or **multiplying/dividing**:

- When measurements are
**added or subtracted**,—the result is the same.**sum**the absolute*or*relative uncertainty - When measurements are
**multiplied or divided**,__sum____the__.*relative*uncertainties

So anytime you **square** a measurement, add the uncertainty **twice**.

To convert between absolute uncertainty and percent uncertainty, we’ll use this formula (*m *= measurement, Δ*u *= absolute uncertainty):

$\overline{){\mathit{m}}{\mathbf{\pm}}{\mathbf{\u2206}}{\mathit{u}}{\mathbf{=}}{\mathit{m}}{\mathbf{\pm}}\mathbf{(}\frac{\mathbf{\u2206}\mathit{u}}{\mathit{m}}\mathbf{\times}\mathbf{100}\mathbf{\%}\mathbf{)}}$

This problem gives us the maximum and minimum values for the length of a cubit. For parts **(a) **and** (b)** of the problem, we just need to **convert the units** from centimeters to meters, write our equation, and **plug in the max and min**. Part **(c)** is a little more complicated: we’ll need the rule for __multiplying measurements with uncertainty.__

We’ll assume that we know the length and diameter of the cylinder to 0.1 cubit—that is, *h *= 9.0 cubits and *d* = 2.0 cubits.

Uncertainty