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

a)
`\delta = 1.4\,\%`, b)
`\delta_(max) = 1.35\,\%`

Explanation:

a) The area formula for a square is:

`A =l^(2)`

The total differential for the area is:

`\Delta A = (\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 = (\Delta A)/(A)* 100\,\%`

`\delta = (1.4\,cm^(2))/(100\,cm^(2)) * 100\,\%`

`\delta = 1.4\,\%`

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

`\Delta A_(max) = \left((\delta)/(100) \right)\cdot A`

`\Delta A_(max) = \left((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)= (\Delta A_(max))/(2\cdot l)`

`\Delta l_(max) = (2.7\,cm^(2))/(2\cdot (10\,cm))`

`\Delta l_(max) = 0.135\,cm`

Lastly, the maximum allowable relative error is:

`\delta_(max) = (\Delta l_(max))/(l)* 100\,\%`

`\delta_(max) = (0.135\,cm)/(10\,cm) * 100\,\%`

`\delta_(max) = 1.35\,\%`