Chapter 6, Problem 43RE

To determine

To find: the total cost of printing newsletters.

Answer to Problem 43RE

The total cost of printing 2500 newsletters is: $230\text{dollars}$

Explanation of Solution

Given information:

It costs a printer $\text{\$}50$ to print 25 copies of a newsletter, after which the marginal cost at $x$ copies is $\frac{dc}{dx}=\frac{2}{\sqrt{x}}\text{dollars per copy}\text{.}$

Calculation:

Given that,

  $\frac{dc}{dx}=\frac{2}{\sqrt{x}}$

Take integration both sides with respect to limits vary from 25 to $x$ and add the cost of printer that is 50 dollars.

  ${{\displaystyle \int }}_{25}^x\frac{dc}{dx}(dx)={{\displaystyle \int }}_{25}^x\frac{2}{\sqrt{x}}dx+50$

If $f$ is continuous at every point of $[a,b],$ and if $F$ is any anti-derivative of $f$ on $[a,b],$ then

  ${{\displaystyle \int }}_a^{b}f(x)dx=F(b)-F(a)$

This part of the fundamental theorem is also called the integral evaluation theorem.

Proof-

Part 1 of the fundamental theorem tells us that an anti-derivative of $f$ exists, namely

  $G(x)={{\displaystyle \int }}_a^{x}f(t)dt$

Thus, if $F$ is any anti-derivative of $f$ then $F(x)=G(x)+C$ for some constant C

  $\begin{array}{l}F(b)-F(a)=[G(b)+C]-[G(a)+C]\\ =G(b)-G(a)\\ {{\displaystyle =\int }}_a^{b}f(t)dt-{{\displaystyle \int }}_a^{b}f(t)dt\\ ={{\displaystyle \int }}_a^{b}f(t)dt-0\\ \Rightarrow F(b)-F(a)={{\displaystyle \int }}_0^{b}f(t)dt\end{array}$

So,

  ${{\displaystyle \int }}_a^{b}f(x)dx=F(b)-F(a)$

Now, use the above proof to find the value of integral.

  $f(x)=\frac{dc}{dx}$

If $F$ is any anti-derivative of $f$ then

  $F(x)=c$

Also, anti-derivative of $\frac{2}{\sqrt{x}}$ is $4\sqrt{x}$

So,

  $\begin{array}{l}c(x)={{\displaystyle \int }}_{25}^x\frac{2}{\sqrt{x}}dx+50\\ =(4\sqrt{x}-4\sqrt{25})+50\\ =4\sqrt{x}+30\end{array}$

Hence, total cost of printing 2500 newsletters is given by-

  $\begin{array}{cc}c(2500)& =4\sqrt{2500}+30\\ & =4\times 50+30\\ & =230\text{dollars}\end{array}$

Hence, total cost of printing 2500 newsletters is: $230\text{dollars}$