How accurately should you measure the edge of a cube to be reasonably sure of calculating the cube’s surface area with an error of no more than 2%?

Verified step by step guidance

1. Understand that the surface area of a cube with edge length \( x \) is given by the formula \( S = 6x^2 \).
2. To find the error in the surface area, we need to use the concept of differentials. The differential \( dS \) represents the change in surface area, and is given by \( dS = \frac{dS}{dx} \cdot dx \).
3. Calculate the derivative of the surface area with respect to \( x \): \( \frac{dS}{dx} = 12x \). This represents how the surface area changes with a small change in \( x \).
4. Set up the inequality for the error: \( \frac{dS}{S} \leq 0.02 \). Substitute \( dS = 12x \cdot dx \) and \( S = 6x^2 \) into the inequality to get \( \frac{12x \cdot dx}{6x^2} \leq 0.02 \).
5. Simplify the inequality to find \( dx \), the allowable error in measuring \( x \): \( \frac{2 \cdot dx}{x} \leq 0.02 \). Solve for \( dx \) to determine the precision needed in measuring the edge length.