Chapter 7.2, Problem 9P

To determine

To find: The original inside diameter of the pipe.

Answer to Problem 9P

The original inside diameter of the pipe is 9.472 mm.

Explanation of Solution

Given:

The thickness of the coating = 1mm.

The reduction in the area is 20%.

Calculation:

The cross-sectional area of the pipe is given by

  $A=\frac{\pi }{4}{d}^2$

Area reduces by 20% and remains 0. 8A. The diameter reduces by 1 mm.

  $\begin{array}{l}A-0.2A=\frac{\pi }{4}{\left(d-1\right)}^2\\ 0.8\left(\frac{\pi }{4}{d}^2\right)=\frac{\pi }{4}{\left(d-1\right)}^2\\ 0.8d^2=\left(d^2+1-2d\right)\\ 0.2d^2-2d+1=0\\ d^2-10d+5=0\end{array}$

The solution of the given equation is obtained by using the quadratic formula.

For a quadratic equation in the form of $a{x}^2+bx+c=0$ , the solution of this equation is given by

  $x_{1,2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Comparing the given equation with the standard quadratic equation, the coefficients are:

  $\begin{array}{l}a=1\\ b=-10\\ c=5\end{array}$

Substituting these in the formula for the roots.

  $\begin{array}{c}d=\frac{-\left(-10\right)\pm \sqrt{{\left(-10\right)}^2-4\times 1\times \left(5\right)}}{2\times 1}\\ =\frac{10\pm \sqrt{100-20}}{2}\\ =\frac{10\pm 8.944}{2}\\ d=9.472\\ d=0.52786\end{array}$

The value of d as 0.52786 is not valid. Therefore, the value of diameter is 9.472 mm.