Chapter 5.3, Problem 67E

To determine

To find: The increase in the cost if the production level is raised from $2000$ yards to $4000$ yards.

Answer to Problem 67E

The increase in the cost if the production level is raised from $2000$ yards to $4000$ yards is equal to $\$58000$ .

Explanation of Solution

Given information: The marginal cost of manufacturing $x$ yards of a certain fabric is $C^{\prime }\left(x\right)=3-0.01x+0.000006x^2$ .

Calculation:

The increase in the cost if the production level is raised from $2000$ yards to $4000$ yards is given by the integral ${\displaystyle \underset{2000}{\overset{4000}{\int }}{C}^{\prime }\left(x\right)dx}={\displaystyle \underset{2000}{\overset{4000}{\int }}\left(3-0.01x+0.000006x^2\right)dx}$ .

Simplify the integral.

  $\begin{array}{c}{\displaystyle \underset{2000}{\overset{4000}{\int }}\left(3-0.01x+0.000006x^2\right)dx}={\displaystyle \underset{2000}{\overset{4000}{\int }}\left(3dx\right)}-{\displaystyle \underset{2000}{\overset{4000}{\int }}0.01xdx}+{\displaystyle \underset{2000}{\overset{4000}{\int }}0.000006x^{2}dx}\\ =3{\displaystyle \underset{2000}{\overset{4000}{\int }}dx}-0.01{\displaystyle \underset{2000}{\overset{4000}{\int }}xdx}+0.000006{\displaystyle \underset{2000}{\overset{4000}{\int }}{x}^{2}dx}\\ =3{\left(x\right)|}_{2000}^{4000}3x-0.01{\left(\frac{x^2}{2}\right)|}_{2000}^{4000}+0.000006{\left(\frac{x^3}{3}\right)|}_{2000}^{4000}\\ =3{\left(x\right)|}_{2000}^{4000}-0.005{\left(\frac{x^2}{2}\right)|}_{2000}^{4000}+0.000002{\left(\frac{x^3}{3}\right)|}_{2000}^{4000}\end{array}$

Further simplify.

  $\begin{array}{c}{\displaystyle \underset{2000}{\overset{4000}{\int }}\left(3-0.01x+0.000006x^2\right)dx}=3\left(4000-2000\right)-0.005\left[{\left(4000\right)}^2-{\left(2000\right)}^2\right]\\ +0.000002\left[{\left(4000\right)}^3-{\left(2000\right)}^3\right]\\ =3\left(2000\right)-0.005\left(16000000-4000000\right)\\ -0.000002\left(64000000000-8000000000\right)\\ =6000-0.005\left(12000000\right)-0.000002\left(58000000000\right)\\ =58000\end{array}$

Therefore, the increase in the cost if the production level is raised from $2000$ yards to $4000$ yards is equal to $\$58000$ .