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The error in measuring the radius of a 5 cm circular rod was 0.2%, If the cross-sectional area of the rod was calculated using this measurement, then the resulting absolute percentage error in the computed area is ______ . (round off to two decimal places)
Correct answer is between '0.39,0.41'. Can you explain this answer?
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The error in measuring the radius of a 5 cm circular rod was 0.2%, If ...
Calculating the Absolute Percentage Error in the Computed Area
To calculate the absolute percentage error in the computed area, we need to first determine the error in the radius measurement and then use it to calculate the error in the area.
Step 1: Determining the Error in the Radius Measurement
Given that the error in measuring the radius of the circular rod is 0.2%, we can calculate the actual error in the radius as follows:
Error in radius = 0.2% of 5 cm = (0.2/100) * 5 cm = 0.01 cm
Step 2: Calculating the Error in the Area
The area of a circle is given by the formula A = πr^2, where A is the area and r is the radius.
To calculate the error in the area, we can differentiate the area formula with respect to the radius and then multiply it by the error in the radius. The formula for the error in the area is given by:
Error in area = 2πr * Error in radius
Substituting the values, we have:
Error in area = 2π * 5 cm * 0.01 cm = 0.1π cm^2
Step 3: Calculating the Absolute Percentage Error
The absolute percentage error is calculated by dividing the error in the area by the computed area and then multiplying by 100.
Absolute percentage error = (Error in area / Computed area) * 100
The computed area is given by:
Computed area = π * (5 cm)^2 = 25π cm^2
Substituting the values, we have:
Absolute percentage error = (0.1π cm^2 / 25π cm^2) * 100 = 0.4%
Conclusion
The resulting absolute percentage error in the computed area is 0.4%. However, we are asked to round off the answer to two decimal places. Since the error is closer to 0.4% than 0.3%, the correct rounded answer is between 0.39 and 0.41, as given.
To calculate the absolute percentage error in the computed area, we need to first determine the error in the radius measurement and then use it to calculate the error in the area.
Step 1: Determining the Error in the Radius Measurement
Given that the error in measuring the radius of the circular rod is 0.2%, we can calculate the actual error in the radius as follows:
Error in radius = 0.2% of 5 cm = (0.2/100) * 5 cm = 0.01 cm
Step 2: Calculating the Error in the Area
The area of a circle is given by the formula A = πr^2, where A is the area and r is the radius.
To calculate the error in the area, we can differentiate the area formula with respect to the radius and then multiply it by the error in the radius. The formula for the error in the area is given by:
Error in area = 2πr * Error in radius
Substituting the values, we have:
Error in area = 2π * 5 cm * 0.01 cm = 0.1π cm^2
Step 3: Calculating the Absolute Percentage Error
The absolute percentage error is calculated by dividing the error in the area by the computed area and then multiplying by 100.
Absolute percentage error = (Error in area / Computed area) * 100
The computed area is given by:
Computed area = π * (5 cm)^2 = 25π cm^2
Substituting the values, we have:
Absolute percentage error = (0.1π cm^2 / 25π cm^2) * 100 = 0.4%
Conclusion
The resulting absolute percentage error in the computed area is 0.4%. However, we are asked to round off the answer to two decimal places. Since the error is closer to 0.4% than 0.3%, the correct rounded answer is between 0.39 and 0.41, as given.
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Community Answer
The error in measuring the radius of a 5 cm circular rod was 0.2%, If ...
A = πr2
For percentage error,
∴
For percentage error,
∴
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The error in measuring the radius of a 5 cm circular rod was 0.2%, If the cross-sectional area of the rod was calculated using this measurement, then the resulting absolute percentage error in the computed area is ______ . (round off to two decimal places)Correct answer is between '0.39,0.41'. Can you explain this answer?
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