Q. 4.117 - Intermediate Algebra

Question

A trailer can carry a maximum weight of $160$ pounds and a maximum volume of $15$ cubic feet. A microwave oven

weighs $30$ pounds and has $2$ cubic feet of volume, while a printer weighs $20$ pounds and has $3$ cubic feet of space.

ⓐ Write a system of inequalities to model this situation.

ⓑ Graph the system.

ⓒ Could $4$ microwaves and $2$ printers be carried on this trailer?

ⓓ Could $7$ microwaves and $3$ printers be carried on this trailer?

Step-by-Step Solution

Answer

a. The system of inequalities is $\left\{\begin{array}{l}30x+20y\le 160\\ 2x+3y\le 15\\ x\ge 0\\ y\ge 0\end{array}\right.$

b. The graph of the system of inequalities is:

c. yes.

d. No.

part (a) Step 1. Form the inequalities

Let

$$\begin{gathered} x=number of microwaves \\ y=number of printers \end{gathered}$$

The weight of microwaves and printers should not be greater than $160$ pounds

$30x+20y\le 160$

The volume of microwaves and printers should not be greater than$15 cubic feet$

$2x+3y\le 15$

The number of microwaves always be greater or equal to zero

$x\ge 0$

The number of printers always be greater or equal to zero

$y\ge 0$

The system of inequalities is:

$\left\{\begin{array}{l}30x+20y\le 160\\ 2x+3y\le 15\\ x\ge 0\\ y\ge 0\end{array}\right.$

Part(b) Step 1. graph the system of inequalities

The graph of the obtained system of inequalities is given as:

part( c) Step 1 . Find 4 , 2 in the solution region

To determine if $4$ microwaves and $2$printers can be carried on this trailer,

we look at the graph to see if the point$\left(4,2\right)$ is in the solution region (dark blue).

Yes $\left(4,2\right)$ is in the solution region so $4$ microwaves and $2$ printers can be carried on this trailer.

part(d) Step 1. Find 7 , 3 in the solution region.

To determine if $7$ microwaves and $3$ printers can be carried on this trailer,

we look at the graph to see if the point $\left(7,3\right)$ is in the solution region (dark blue).

No, it is not in the solution region.

So, $7$ microwaves and $3$ printers cannot be carried on the trailer.