Q. 4.120 - Intermediate Algebra

Question

Philip’s doctor tells him he should add at least $1,000$ more calories per day to his usual diet. Philip wants to buy

protein bars that cost $\backslash (1.80$each and have 140 calories and juice that costs $\backslash )1.25$ per bottle and have $125$ calories.

He doesn’t want to spend more than $\$12$.

ⓐ Write a system of inequalities that models this situation.

ⓑ Graph the system.

ⓒ Can he buy $3$ protein bars and $5$ bottles of juice?

ⓓ Can he buy $5$ protein bars and $3$ bottles of juice?

Step-by-Step Solution

Answer

Part (a). The obtained system of inequalities is:

$\left\{\begin{array}{l}140x+125y\ge 1000\\ 1.80x+1.25y\le 12\\ x\ge 0\\ y\ge 0\end{array}\right.$

Part (b). The graph of the system of inequalities is:

Part (c). Yes

Part (d). No

Part (a) Step 1. Form the inequalities.

Let  

$$\begin{gathered} x=number of protein bars \\ y=number ofbottles of juice \end{gathered}$$

Philip's has to consume at least $1000$calories where each protein jar has $140$ calories and a bottle of juice has $125$ calories. So,

$140x+125y\ge 1000$ 

$\$1.80$ for protein jar and $\$1.25$ for a bottle of protein must not be more than $\$12$. So,

$1.80x+1.25y\le 12$

The number of protein bars is equal or greater than zero. So,

$x\ge 0$

The number of bottles is equal or greater than zero.

$y\ge 0$

The obtained system of inequalities

$\left\{\begin{array}{l}140x+125y\ge 1000\\ 1.80x+1.25y\le 12\\ x\ge 0\\ y\ge 0\end{array}\right.$

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

The graph for the obtained system of inequalities is:  

Part (c) Step 1. Find 3 , 5 in the solution region

To determine if he can buy $3$ protein bars and $5$ bottles of juice,  we look at the graph to see if the point $\left(3,5\right)$is in the solution region (dark blue).

Yes, it is in the solution region. So Philip can buy $3$ protein bars and $5$ bottles of juice.

Part (d) step 1. Find 5 , 3 in the solution region.

o determine if he can buy $5$ protein bars and $3$ bottles of juice,  we look at the graph to see if the point  $\left(5,3\right)$ is in the solution region (dark blue).

No, it is not in the solution region. So Philip cannot buy $5$ protein bars and $3$ bottles of juices.