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The lengths of two components will be measured several
Chapter 3, Problem 20E(choose chapter or problem)
The lengths of two components will be measured several times. The uncertainty in each measurement of the length of the first component is \(\sigma_1=0.02~\mathrm {cm}\), and the uncertainty in each measurement of the length of the second component is \(\sigma_2=0.08~\mathrm {cm}\). Let \(\overline X\) denote the average of the measurements of the first component, and let \(\overline Y\) denote the average of the measurements of the second component. The total length of the two components will be estimated with the quantity \(\overline X + \overline Y\).
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.
Equation Transcription:
Text Transcription:
sigma_1=0.02 cm
sigma_2=0.08 cm
overline{X}
overline{Y}
overline{X}+overline{Y}
16-n
Questions & Answers
QUESTION:
The lengths of two components will be measured several times. The uncertainty in each measurement of the length of the first component is \(\sigma_1=0.02~\mathrm {cm}\), and the uncertainty in each measurement of the length of the second component is \(\sigma_2=0.08~\mathrm {cm}\). Let \(\overline X\) denote the average of the measurements of the first component, and let \(\overline Y\) denote the average of the measurements of the second component. The total length of the two components will be estimated with the quantity \(\overline X + \overline Y\).
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.
Equation Transcription:
Text Transcription:
sigma_1=0.02 cm
sigma_2=0.08 cm
overline{X}
overline{Y}
overline{X}+overline{Y}
16-n
ANSWER:
Answer
Step 1 of 4
Uncertainty means standard deviation of the measurements
The uncertainty in the length of the first component is
The uncertainty in the length of the second component is
Let denote the average of the measurements of the first component
Let denote the average of the measurements of the second component
The total length of the components will be estimated with the quantity