Question

The side of a square is measured to be 12 inches, with a possible error of \(\frac{1}{64}\) inch. Use differentials to approximate the possible error and the relative error in computing the area of the square.

Short answer

The possible error in computing the area of the square is approximately \( \frac{3}{8} \) square inches, and the relative error is approximately \( \frac{1}{384} \) or 0.0026 in decimal notation.

Step-by-step solution

Define the Original Function

First, define the given function. The area \(A\) of a square is calculated as the side length \(s\) squared, that is \(A = s^2\). In this case, the square has a side length of 12 inches.

Compute Differential

Next, compute the differential of the area \(dA\). Since \(A = s^2\), the differential \(dA\) can be calculated as \(dA = 2s ds\). Substituting \(s = 12\) inches and \(ds = \frac{1}{64}\) inch (as the possible error of the side length) into the equation gives \(dA = 2(12) * \frac{1}{64} = \frac{3}{8}\) square inches.

Compute Relative Error

Finally, compute the relative error in calculating the area. Relative error is defined as the absolute error divided by the correct value. In this case, the absolute error is the differential of the area \(dA = \frac{3}{8}\) square inches and the correct value is the actual area of the square \(A = 12^2 = 144\) square inches. Therefore, the relative error is \(\frac{dA}{A} = \frac{\frac{3}{8}}{144} = \frac{1}{384}\) or approximately 0.0026 when converted to a decimal.