Chapter 8.4, Problem 12PPS

a.

To determine

To find:

The function for the average cost of a T-shirt sold and to graph.

Answer to Problem 12PPS

The value is $f\left(n\right)=\frac{9.5n+217.5}{n}$ .

Explanation of Solution

Given information:

Each T-shirts costs $\$9.50$ and there is set up fee of $\$75$ .

Let n be the number of T-shirts ordered.

There is a set-up fee of $\$75$ .

  $15$ T-shirts that cost $\$9.50$ which costs a total of $\$142.50$ , will be given for free to the council members.

Thus, this mean that the total cost for n T-shirts will be:

  $\begin{array}{l}=9.50\left(n\right)+75+142.50\\ =9.5+217.50\end{array}$

Therefore, the average cost of ordering n T-shirts is:

  $=\frac{9.50n+217.5}{n}$

Hence, the function that gives the average cost each T-shirts is:

  $f\left(n\right)=\frac{9.50n+217.5}{n}$

Graph:

  

Interpretation:

$n$	$f\left(n\right)=\frac{9.50n+217.5}{n}$
$1$	$227$
$3$	$82$
$15$	$24$
$30$	$16.75$
$58$	$13.25$
$87$	$12$

Plot each ordered pair and connect the points using a smooth curve. Hence, the function has $x=0$ and $y=0$ as asymptotes.

b.

To determine

To Find:

The average cost if $200$ shirts are ordered.

Answer to Problem 12PPS

The average cost is $\$10.68$ and for 500 shirts $\$9.95$ .

Explanation of Solution

Given information:

Each T-shirts costs $\$9.50$ and there is set up fee of $\$75$ .

Let t be the number of ordered T-shirts.

The average cost $c\left(t\right)$ is

  $\begin{array}{l}c\left(t\right)=\frac{9.50t+75}{t-15}\\ c\left(200\right)=\frac{9.50\times 200+75}{200-15}\\ c\left(200\right)=\frac{1975}{185}\approx 10.68\end{array}$

In case 500 shirts are ordered,

  $\begin{array}{l}c\left(t\right)=\frac{9.50t+75}{t-15}\\ c\left(500\right)=\frac{9.50\times 500+75}{500-15}\\ c\left(500\right)=\frac{4825}{485}\approx 9.95\end{array}$

Therefore, the average cost is $\$10.68$ and for 500 shirts $\$9.95$ .

c.

To determine

To find:

The number of T-shirts must be ordered to bring the average cost under $\$9.75$ .

Answer to Problem 12PPS

The approximately the value is $t>885$ T-shirts.

Explanation of Solution

Given information:

Each T-shirts costs $\$9.50$ and there is set up fee of $\$75$ .

The average cost $c\left(t\right)$ is

  $\begin{array}{l}c\left(t\right)=\frac{9.50t+75}{t-15}\\ c\left(t\right)<9.75\\ \frac{9.50t+75}{t-15}<9.75\\ \frac{9.50t+75}{t-15}-9.75<0\\ \frac{9.50t+75-9.75\left(t-15\right)}{t-15}<0\\ \frac{9.50t+75-9.75t+146.25}{t-15}<0\\ \frac{-0.25t+221.25}{t-15}<0\\ \left(0.25t-221.25\right)\left(t-15\right)>0\\ 0.25\left(t-885\right)\left(t-15\right)<0\\ \left(t-885\right)\left(t-15\right)<0\\ \Rightarrow t\in \left(-\infty ,15\right)\cup \left(885,\infty \right)\end{array}$

Therefore, $t>15$ . Hence the result is $t>885$ .

Hence, the approximately the value is $t>885$ T-shirts.