Problem 25 - Calculus of a Single Variable

Question

Area The measurement of the side of a square floor tile is 10 inches, with a possible error of \(\frac{1}{32}\) inch. (a) Use differentials to approximate the possible propagated error in computing the area of the square. (b) Approximate the percent error in computing the area of the square.

Step-by-Step Solution

Answer

The possible propagated error in computing the area of the square is approximately 0.625 square inches. The approximate percent error in computing the area of the square is 0.62%.

Step 1: Formulating the function

Firstly, we need to express the area \(A\) of the square as a function of the length of its side \(x\). Since a square has all sides equal, its area is given by \(A = x^2\).

Step 2: Finding the differential

Next, we find the differential \(dA\), which represents the error in the area. The differential of a function is obtained by differentiating the function and multiplying by the differential of the variable. Thus, \(dA = 2x \cdot dx \). Here, \(dx\) is the error in the measurement of the side of the square, i.e., \(\frac{1}{32}\) inch.

Step 3: Calculating the Error in Area

Now that we have found the differential, we can substitute the given values into it to find the maximum propagated error in the area. Substituting \(x=10\) inches and \(dx= \frac{1}{32}\) inch into the equation, we get \(dA = 2(10)(\frac{1}{32}) = 0.625\) square inches.

Step 4: Finding the Percent Error in Area

Finally, we find the percent error by dividing the error by the original quantity (which is the area of the square) and multiplying by 100. This gives us \(\frac{dA}{A} \times 100 = \frac{0.625}{10^2} \times 100 = 0.625% \) We round to the standard number of significant digits in percent error problems, which is typically two or three. In this case, rounding to two decimal places, our final answer for percent error is 0.62%.