Answered: A 2-inch schedule 40 (2.067 in.) pipe…

A 2-inch schedule 40 (2.067 in.) pipe showed a deviation of 0.113 cm below the true value of its inside diameter. What is the deviation of the inside area?

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Question

A 2-inch schedule 40 (2.067 in.) pipe showed a deviation of 0.113 cm below the true value of its inside diameter. What is the deviation of the inside area?

Expert Solution

Step 1

The diameter (d) of the inside be defined as 2.067 inches. Therefore, the radius (r) be given as,

$\begin{array}{rcl} r & = & \frac{d}{2} \\ = & \frac{2.067 inch}{2} \\ = & 1.0335 inch \\ = & (1.0335 inch) (\frac{2.54 cm}{1 inch}) \\ = & 2.625 cm \end{array}$

Step 2

Let A be defined as the area. Therefore,

$\begin{array}{rcl} A & = & {πr}^{2} \\ Taking \log both side, \\ \log (A) & = & \log (π r^{2}) \\ \log (A) & = & \log (π) + 2 \log (r) \\ On Differentiating, \\ \frac{d A}{A} & = & \frac{2}{r} d r \\ d A & = & \frac{2 A}{r} d r \\ d A & = & \frac{2 π r^{2}}{r} d r \\ d A & = & 2 π r d r \end{array}$

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