Q. 61 - Precalculus Enhanced with Graphing Utilities

Question

Transporting Goods A small truck can carry no more than 1600 pounds (lb) of cargo nor more than 150 cubic $\left({\mathrm{ft}}^3\right)$ of cargo. A printer weighs 20 lb and occupies 3 ${\mathrm{ft}}^3$ of space. A microwave oven weighs 30 lb and occupies 2 ${\mathrm{ft}}^3$ of space.

(a) Using x to represent the number of microwave ovens and y to represent the number of printers, write a system of linear inequalities that describes the number of ovens and printers that can be hauled by the truck. 

(b) Graph the system and label the corner points.

Step-by-Step Solution

Answer

$$\begin{gathered} \left(a\right) \left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ 3x+2y\le 160\\ 2x+3y\le 150\end{array}\right. \\ \left(b\right) \mathrm{The} \mathrm{corner} \mathrm{points} \mathrm{are} (0,0),(0,50),(36,26) \mathrm{and} (53.33,0). \end{gathered}$$

Part (a) Step 1. Given information

The goal of the task is to write a system of linear inequalities that describes the number of ovens and printers that a truck can pull.

x- represents the number of microwave ovens

y- represents the number of printers

Part (a) Step 2. Make a table

	pounds (lb)	cubic feet
x	30	2
y	20	3

Part (a) Step 3. Find the system of linear inequalities

A small truck can carry no more than $1600$ pounds :

$30x+20y\le 1600$

Divide it by $10$.

$3x+2y\le 160$

The truck can carry a maximum of $150$ cubic feet of cargo:

$2x+3y\le 150$

The truck can tow ovens and printers.

So, $x\ge 0,y\ge 0$

The system of linear inequalities is : 

$\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ 3x+2y\le 160\\ 2x+3y\le 150\end{array}\right.$

Part (b) Step 1. Given information

The goal of the problem is to graph the system of linear inequalities.

$\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ 3x+2y\le 160\\ 2x+3y\le 150\end{array}\right.$

Part (b) Step 2. Graph the system

Part (b) Step 3. Find coordinates of point A

Solving the system of equations 

$$\begin{gathered} 3x+2y=160 \\ 2x+3y=150 \end{gathered}$$

The point A will be $(36,26)$.

Part (b) Step 4. Find coordinates of point B

Solving the system of equations 

$$\begin{gathered} y=0 \\ 3x+2y=160 \end{gathered}$$

The point B will be $(53.33,0)$.