The lengths of two components will be measured several times. The uncertainty in each measurement of the length of the first component is σ1 = 0.02 cm, and the uncertainty in each measurement of the length of the second component is σ₂ = 0.08 cm. Let X̅ denote the average of the measurements of the first component, and let Y̅ denote the average of the measurements of the second component. The total length of the two components will be estimated with the quantity X̅ + Y̅.

a. Find the uncertainty in the total length if the first component is measured 4 times and the second component is measured 12 times.

b. Find the uncertainty in the total length in terms of n if the first component is measured n times and the second component is measured 16 − n times.

c. Determine the best way to allocate 16 measurements between the components by determining the value of n that minimizes the uncertainty.