Chapter 5.5, Problem 45E

To determine

To calculate the accuracy you should maintain in measure the side of the square.

Answer to Problem 45E

You should measure x with an error dx that is no more than 1% of the true value.

Explanation of Solution

Given information:

Be sure of calculating the area to within 2% of its true value.

Calculation :

Consider the side of the square be x .

You want any accuracy in your measurement to be small enough to make the corresponding increment $\Delta S$ in the surface area satisfy the inequality.

  $\left|\Delta S\right|\le \frac{2}{100}S=\frac{2x^2}{100}$

Replace $\Delta S$ in this inequality by its approximation

  $\begin{array}{l}dS=\left(\frac{dS}{dx}\right)dx\\ =2xdx\end{array}$

This gives

  $\begin{array}{l}\left|2xdx\right|\le \frac{2x^2}{100}\\ or,\left|dx\right|\le \frac{1}{2x}\cdot \frac{2x^2}{100}\\ =\frac{x}{100}\\ =0.01x\end{array}$

You should measure x with an error dx that is no more than 1% of the true value.