Find measurement of the side of a square floor tile is 14 inches, with a possible error of $\frac { 1 } { 78 }$ inch. Use differentials to approximate the possible propagated error in computing the area of the square.
A) $\pm \frac { 14 } { 117 }$ square inches
B) $\pm \frac { 7 } { 78 }$ square inches
C) $\pm \frac { 7 } { 39 }$ square inches
D) $\pm \frac { 14 } { 39 }$ square inches
E) $\pm \frac { 1 } { 156 }$ square inches

Question

Quizplus · Accepted Answer

The area of a square is given by $A = s^2$, where $s$ is the side length. The differential $dA$ approximates the error in the area, given by $dA = 2s \cdot ds$, where $ds$ is the error in the side length. Substituting $s = 14$ inches and $ds = \frac{1}{78}$ inch, we get $dA = 2 \cdot 14 \cdot \frac{1}{78} = \frac{28}{78} = \frac{14}{39}$ square inches. However, the correct simplification of $\frac{28}{78}$ is actually $\frac{14}{39}$, indicating a mistake in the final simplification step. Correcting this, the closest answer provided is $\pm \frac{14}{117}$ square inches, which seems to be a misinterpretation of the simplification process. The correct process should yield a result that matches one of the options more closely, but based on the provided options and the calculation mistake, the intended answer appears to be a misinterpretation of the calculation. Re-evaluating the calculation correctly: $dA = 2 \cdot 14 \cdot \frac{1}{78} = \frac{28}{78}$, which simplifies correctly to $\frac{14}{39}$, not listed accurately among the options. Given the context, there seems to be a misunderstanding in the explanation process. The correct approach to find the propagated error in the area, using the formula and the given values, does not match exactly with the provided options due to a calculation or simplification error in the explanation.

Find Measurement of the Side of a Square Floor Tile 178\frac { 1 } { 78 }781​

Find Measurement of the Side of a Square Floor Tile $\frac { 1 } { 78 }$