Physics quiz question: discrepancy between acre areas

Physics 205A Quiz 1, fall semester 2014
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Comprehensive Problem 1.57

Evaluate the following calculation, using an appropriate number of significant figures and/or decimal places. (1 ft = 12 in.)

4.3560×10⁴ ft² – (208 ft + 9 in)² = ?

(A) 0 ft².
(B) –17 ft².
(C) –16.6 ft².
(D) –16.563 ft².

Correct answer (highlight to unhide): (A)

The relevant conversion is 1 ft = 12 in. Thus the second term can be expressed as:

(208 ft + 9 in)² = (208 ft + 9 in·(1 ft/12 in))² = (208 ft + 0.75 ft)²,

where the digits in green are significant (and thus limit the resulting numbers of significant figures or decimal places in each calculation), and the digits in yellow are not significant (and thus must be rounded at the end of the calculation, to avoid cumulative round-off errors). Thus the second term (with the addition of a value with no significant decimal places with a value with one significant decimal place) becomes:

(208 ft + 0.75 ft)² = (208.75 ft)²,

which has no significant decimal places, and three significant figures. Squaring this term results in a value with three significant figures:

(208.75 ft)² = 4.35765625×10⁴ ft².

Subtracting this value (which only has three significant figures, or two significant decimal places) from the first term (which has five significant figures, or four significant decimal places) results in a value which should only have two significant decimal places when also expressed with the same power of ten in scientific notation:

4.3560×10⁴ ft² – 4.35765625×10⁴ ft² = –0.00165625×10⁴ ft².

Thus rounded to only two significant decimal places, the answer is 0.

(What would happen if the two values to be subtracted from each other were not expressed in scientific notation? That is:

43,560 ft² – 43,576.5625 ft² = ?

Expressed this way, the first term has five significant figures, or zero significant decimal places. The second term has three significant figures (as before), but expressed in this manner not only does it not have any significant decimal places, but the ones and tens values to the left of the decimal place are also not significant. The resulting subtraction:

43,560 ft² – 43,576.5625 ft² = –16.5626 ft²,

must also reflect that (ones and tens values to the left of the decimal place not being significant), such that not only the "5626" decimal digits are not significant, but also the "16" digits are not significant as well, and thus the result of this calculation is 0, rounded to the nearest hundreds place (which is where values become significant).)

Sections 70854, 70855, 73320
Exam code: quiz01w34K
(A) : 3 students
(B) : 25 students
(C) : 21 students
(D) : 26 students

Success level: 4%
Discrimination index (Aubrecht & Aubrecht, 1983): 0