You are looking to purchase a small piece of land in Hong...

You are looking to purchase a small piece of land in Hong Kong. The price is "only" \$60,000 $ per square meter! The land title says the dimensions are $20 \mathrm{m} \times 30 \mathrm{m} $By how much would the total price change if you measured the parcel with a steel tape measure on a day when the temperature was $20^{\circ} \mathrm{C}$ above normal?

Key Concepts

Thermal Expansion

Thermal expansion refers to the physical phenomenon where materials change their dimensions in response to changes in temperature. In the context of measuring with a steel tape measure, the material’s length increases as the temperature rises, which means that any measurement taken under these conditions may be slightly inaccurate when compared to the standard at which the tape was calibrated. This concept is fundamental in materials science and engineering because it affects precise measurements and tolerances.

Measurement Error and Instrument Calibration

Measurement error arises when there is a discrepancy between the actual value and the measured value due to various factors, including the calibration of the instrument and environmental influences like temperature. In this problem, the error induced by the thermal expansion of the steel tape measure leads to a systematic error in determining the dimensions of the property, and consequently, the computed area and total cost. Understanding measurement error is essential in quality assurance and experimental design.

Error Propagation in Area Calculations

Error propagation refers to the process by which uncertainties in measured quantities affect the uncertainty in a result that is derived from those measurements. In the context of determining the area of a land parcel, any small error in the linear measurements (width and length) due to thermal expansion becomes amplified when these measurements are multiplied to compute the area. This concept is crucial in fields like metrology and economics where slight measurement discrepancies can lead to significant financial implications.

Transcript

00:01 Change in temperature causes a change in length.

00:02 To find the date change, we apply the linear thermal expansion formula.

00:08 There is, delta l is equal to alpha times l times delta t, right? and thermal coefficient of expansion of steel is alpha is equal to 12 multiplied by 10 to the power minus 6 degree centigrade per degree centigrade.

00:31 When the temperature is normal, the area measured by steel would be, so area 1 is equal to 20 multiplied by 30, that is equal to 600 meter square, which make the total price of date piece of land, price, price 1 is equal to 600 multiplied by 60 ,000, that is equal to 36 multiplied by 10 to the power 6 dollars right well on the hotter day when the temperature increases by 20 degrees and degrade the steel tape major length will increase due to the increase of temperature the tape measure will now yield shorter dimensions for this piece of length since it expanded but the dimensions of the land is still the same well to determine the short dimensions the tape measure will yield we must calculate the change of length in the tape measure in each of the dimension separately right so the change in the 20 meter dimension would be so the change in 20 meter dimension would be delta l is equal to 12 times 10 to the power six minus six multiple 20 multiplied by 20 that is equal to 4 .8 multiply by 10 to the power minus 3 10 to the power minus 3 meter thus the tip measure will yield x is equal to x is equal to 20 minus 4 .8 times 10 to the power minus 3 and that is equal to 19 .9 952 meter instead of 20 meter so instead of 20 meter well on the other hand the change in a third meter dimension would be so change in 30 meter dimension so change in 30 meter dimension would be delta l is equal to 7 .2 times 10 to 3 meter thus the tape measure will yield y is equal to y is equal to 30 minus 7 .2 times 10 to the power minus 3 that is equal to 29 .9928 meter instead of 30 meter now area 2 is equal to x multiplied y and x multiply by y is equal to 599 .712346 meter square...

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