Problem 16
Question
Exercises \(13-16\) ask a variety of questions dealing with approximating error and sensitivity analysis. It is "common sense" that it is far better to measure a long distance with a long measuring tape rather than a short one. A measured distance \(D\) can be viewed as the product of the length \(\ell\) of a measuring tape times the number \(n\) of times it was used. For instance, using a \(3^{\prime}\) tape 10 times gives a length of \(30^{\prime}\). To measure the same distance with a \(12^{\prime}\) tape, we would use the tape 2.5 times. (l.e., \(30=12 \times 2.5 .\) Thus \(D=n \ell\). Suppose each time a measurement is taken with the tape, the recorded distance is within \(1 / 16 "\) of the actual distance. (l.e., \(d \ell=1 / 16^{\prime \prime} \approx 0.00 \mathrm{fft}\). Using differentials, show why common sense proves correct in that it is better to use a long tape to measure long distances.
Step-by-Step Solution
Verified Answer
Using a longer tape results in less error for measuring the same distance.
1Step 1: Understand the Error in Measurement
When a measurement is taken with a tape, the recorded distance may deviate from the actual distance by ±\(\delta \ell\), where \(\delta \ell = \frac{1}{16}\) inches. We need to calculate the total error in the measurement of distance \(D\).
2Step 2: Set up the Differential Equation for Total Error
The measured distance \(D\) is given by \(D = n \ell\), where \(n\) is the number of times the tape is used and \(\ell\) is the length of the tape. The differential for \(D\), denoted as \(dD\), represents the error propagated in \(D\) and can be calculated using \(dD = n \, d\ell\), where \(d\ell\) is the measurement error per use.
3Step 3: Calculate the Total Error for Different Tape Lengths
For a short tape, say \(\ell = 3\) ft, the number of uses is \(n = \frac{D}{3}\) and the error is \(dD = \frac{D}{3} \, d\ell\). For a longer tape, say \(\ell = 12\) ft, the number of uses is \(n = \frac{D}{12}\) and the error is \(dD = \frac{D}{12} \, d\ell\). As \(\ell\) increases, \(dD\) decreases, showing that the total error is less for longer tape lengths.
4Step 4: Compare Error Values
Substitute \(d\ell = \frac{1}{16} \) inches (converted to feet) into the equations for both short and long tapes. For \(\ell = 3\) ft, the total error is greater than for \(\ell = 12\) ft given the same \(D\). This shows the difference in error magnitude.
Key Concepts
Understanding Error ApproximationThe Role of Sensitivity AnalysisImpact of Measurement Error
Understanding Error Approximation
Error approximation helps us gauge how far our measurements might be from the true values. When we use a measuring tape, the actual distance could vary slightly each time we record it. This deviation is known as the measurement error, which is \(\delta \ell = \frac{1}{16}\) inches in our scenario.
To understand this better, consider the equation for the total distance: \(D = n \ell\), where \(n\) is the number of times the tape measure is used and \(\ell\) is the tape's length. The error propagation in \(D\), denoted as \(dD\), can be approximated using differentials. This gives us the formula \(dD = n \, d\ell\), where \(d\ell\) represents the error per use of the tape.
By using different tape lengths and computing \(dD\) for each, you can understand how the choice of tape length impacts the total measurement error. Longer tapes will result in fewer overall measurements (smaller \(n\)), thus reducing cumulative errors and making the long tapes more accurate.
To understand this better, consider the equation for the total distance: \(D = n \ell\), where \(n\) is the number of times the tape measure is used and \(\ell\) is the tape's length. The error propagation in \(D\), denoted as \(dD\), can be approximated using differentials. This gives us the formula \(dD = n \, d\ell\), where \(d\ell\) represents the error per use of the tape.
By using different tape lengths and computing \(dD\) for each, you can understand how the choice of tape length impacts the total measurement error. Longer tapes will result in fewer overall measurements (smaller \(n\)), thus reducing cumulative errors and making the long tapes more accurate.
The Role of Sensitivity Analysis
Sensitivity analysis in differential calculus examines how the output of a function responds to changes in its inputs. In the context of measurement, it lets us evaluate how different lengths of the measuring tape affect the total measurement error.
The equation \(dD = n \, d\ell\) shows that \(dD\), the error in distance, is influenced directly by both \(n\) and \(d\ell\). When \(\ell = 3\) feet, \(n\) is greater compared to when \(\ell = 12\) feet, making the process more sensitive to error since \(n = \frac{D}{\ell}\).
By comparing outcomes for different tape lengths, sensitivity analysis confirms the common sense that longer tapes create less sensitivity to measurement errors. This analysis is crucial for ensuring accuracy, especially in fields requiring precise measurements, such as construction and engineering.
The equation \(dD = n \, d\ell\) shows that \(dD\), the error in distance, is influenced directly by both \(n\) and \(d\ell\). When \(\ell = 3\) feet, \(n\) is greater compared to when \(\ell = 12\) feet, making the process more sensitive to error since \(n = \frac{D}{\ell}\).
By comparing outcomes for different tape lengths, sensitivity analysis confirms the common sense that longer tapes create less sensitivity to measurement errors. This analysis is crucial for ensuring accuracy, especially in fields requiring precise measurements, such as construction and engineering.
Impact of Measurement Error
Measurement error is crucial in determining the accuracy of our measurements. Each time the tape measure is used, \(d\ell = \frac{1}{16}\) inches error can accumulate if the measuring process is repeated many times. This error impacts the computation of the total measured distance, \(D\).
For a short tape, such as \(\ell = 3\) feet, the total error calculated is more because it involves more repetitions of the measuring process (higher \(n\)). Specifically, \(dD = \frac{D}{3}\, d\ell\) becomes notable for longer distances. In contrast, a 12-foot tape requires fewer repetitions, resulting in less total error: \(dD = \frac{D}{12}\, d\ell\).
This difference demonstrates how using a longer measuring tape reduces cumulative error, reinforcing the idea that a larger \(\ell\) leads to less measurement error, supporting more reliable and accurate measurements.
For a short tape, such as \(\ell = 3\) feet, the total error calculated is more because it involves more repetitions of the measuring process (higher \(n\)). Specifically, \(dD = \frac{D}{3}\, d\ell\) becomes notable for longer distances. In contrast, a 12-foot tape requires fewer repetitions, resulting in less total error: \(dD = \frac{D}{12}\, d\ell\).
This difference demonstrates how using a longer measuring tape reduces cumulative error, reinforcing the idea that a larger \(\ell\) leads to less measurement error, supporting more reliable and accurate measurements.
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