Question

The measurement of a side of a square is found to be 10 centimeters, with a possible error of 0.07 centimeter. (a) Approximate the percent error in computing the area of the square. % (b) Estimate the maximum allowable percent error in measuring the side if the error in computing the area cannot exceed 2.7%. %

Answer 1

Answer:

a) $\delta = 1.4\,\%$, b) $\delta_{max} = 1.35\,\%$

Step-by-step explanation:

a) The area formula for a square is:

$A =l^{2}$

The total differential for the area is:

$\Delta A = \frac{\partial A}{\partial l}\cdot \Delta l$

$\Delta A = 2\cdot l \cdot \Delta l$

The absolute error for the area of the square is:

$\Delta A = 2\cdot (10\,cm)\cdot (0.07\,cm)$

$\Delta A = 1.4\,cm^{2}$

Thus, the relative error is:

$\delta = \frac{\Delta A}{A}\times 100\,\%$

$\delta = \frac{1.4\,cm^{2}}{100\,cm^{2}} \times 100\,\%$

$\delta = 1.4\,\%$

b) The maximum allowable absolute error for the area of the square is:

$\Delta A_{max} = \left(\frac{\delta}{100} \right)\cdot A$

$\Delta A_{max} = \left(\frac{2.7}{100} \right)\cdot (100\,cm^{2})$

$\Delta A_{max} = 2.7\,cm^{2}$

The maximum allowable absolute error for the length of a side of the square is:

$\Delta l_{max}= \frac{\Delta A_{max}}{2\cdot l}$

$\Delta l_{max} = \frac{2.7\,cm^{2}}{2\cdot (10\,cm)}$

$\Delta l_{max} = 0.135\,cm$

Lastly, the maximum allowable relative error is:

$\delta_{max} = \frac{\Delta l_{max}}{l}\times 100\,\%$

$\delta_{max} = \frac{0.135\,cm}{10\,cm} \times 100\,\%$

$\delta_{max} = 1.35\,\%$