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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%. %

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Answer:

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\,\%

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