Chapter 4.7, Problem 43PPS

(a)

To determine

To Draw:

A table organizing the given information with columns showing the total order prize for each size order.

Explanation of Solution

Given Information:

A basketball team is ordering T-shirts with the team logo on the front and the players’ names on the back.A graphic design store charges $10 to set up the artwork plus $10 per shirt,$4 each for the team logo , and $2 to print the last name for an order of 10 shirts or less.For orders of 11-20 shirts, a 5% discount is given.For orders of more than 20 shirts, a 10% discount is given.

Calculation:

Let

x = Number of shirts ordered

Fixed cost :

Cost for setup = $10

Variable cost:

For 1 to 10 shirts ; Cost per shirt is $10 , Cost per logo is $4 , Cost per last name is $2. So, total Variable cost is $16 per shirt.

For 11 to 20 shirts ; 5% discount

For 21 and more shirts ; 10% discount

Summarize this information in a table :

Number of shirts ordered (x)	Total Prize (f(x))
$1\le x\le 10$	$10+(10+4+2)x=10+16x$
$10<x\le 20$	$\begin{array}{l}(10+16x)100\%-(10+16x)5\%=(10+16x)95\%=(10+16x)(0.95)\\ =9.5+15.20x\end{array}$
$x>20$	$\begin{array}{l}(10+16x)100\%-(10+16x)10\%=(10+16x)90\%=(10+16x)(0.90)\\ =9+14.40x\end{array}$

(b)

To determine

To Find :

Write an equation representing the total price for an order of x shirts.

Answer to Problem 43PPS

  $y=f(x)=\{\begin{array}{l}10+16x;1\le x\le 10\\ 9.50+15.20x;10<x\le 20\\ 9+14.40x;20<x\end{array}$

Explanation of Solution

Given Information:

From part (a)

Number of shirts ordered (x)	Total Prize (f(x))
$1\le x\le 10$	$10+(10+4+2)x=10+16x$
$10<x\le 20$	$\begin{array}{l}(10+16x)100\%-(10+16x)5\%=(10+16x)95\%=(10+16x)(0.95)\\ =9.5+15.20x\end{array}$
$x>20$	$\begin{array}{l}(10+16x)100\%-(10+16x)10\%=(10+16x)90\%=(10+16x)(0.90)\\ =9+14.40x\end{array}$

Calculation:

From the above table , the equation for the total price ( f(x)) for an order of x shirts is given by:

  $y=f(x)=\{\begin{array}{l}10+16x;1\le x\le 10\\ 9.50+15.20x;10<x\le 20\\ 9+14.40x;20<x\end{array}$

(c)

To determine

To Graph :

The piece-wise relation.

Explanation of Solution

Given Information:

From part (b)

The equation for the total price ( f(x)) for an order of x shirts is given by:

  $y=f(x)=\{\begin{array}{l}10+16x;1\le x\le 10\\ 9.50+15.20x;10<x\le 20\\ 9+14.40x;20<x\end{array}$

Calculation:

  $y=f(x)=\{\begin{array}{l}10+16x;1\le x\le 10\\ 9.50+15.20x;10<x\le 20\\ 9+14.40x;20<x\end{array}$

Make a table for different values of x :

x	$y=f(x)=\{\begin{array}{l}10+16x;1\le x\le 10\\ 9.50+15.20x;10<x\le 20\\ 9+14.40x;20<x\end{array}$
1	26
10	170
11	176.70
20	313.50
21	311.40
25	369

Plot the points and join the line :