Q 323 - Intermediate Algebra

Question

Mark is attempting to build muscle mass and so he needs to eat more than an additional $80$ grams of protein a day. A bottle of protein water costs \($3.20$ and a protein bar costs \)$1.75$. The protein water supplies $27$ grams of protein and the bar supplies $16$ gram. If he has $$10$ dollars to spend

ⓐ Write a system of inequalities to model this situation. 

ⓑ Graph the system.

ⓒCould he buy $3$ bottles of protein water and $1$ protein bar?

ⓓ Could he buy no bottles of protein water and $5$ protein bars? 

Step-by-Step Solution

Answer

Part a. A system of inequalities to model the situation is $\left\{\begin{array}{l}27x+16y\ge 80\\ 3.20x+1.75y\le 10\\ x\ge 0\\ y\ge 0\end{array}\right.$

Part b. The graph of the system is

Part c. No, he cannot buy $3$ bottles of protein water and $1$ protein bar

Part d. Yes, he can buy no bottles of protein water and $5$ protein bars. 

Part (a) Step 1. Form the system of inequalities

Let $x$ represents the number of protein water bought and $y$ represents the number of protein bars bought.

Now one protein water supplies $27$ grams of protein and one protein bar supplies $16$ grams. In total he needs to consume at least $80$ grams of protein, so the inequality can be written as

$27x+16y\ge 80$

The price of one protein water is $$3.20$ and price of of one protein bar is $$1.75$. He has only $$10$ to spend, so the inequality can be written as

$3.20x+1.75y\le 10$

Now number of protein bars and protein water will always be a non-negative quantity, so the system of inequalities is  

style="max-width: none; vertical-align: -43px;" $\left\{\begin{array}{l}27x+16y\ge 80\\ 3.20x+1.75y\le 10\\ x\ge 0\\ y\ge 0\end{array}\right.$

Part (b) Step 1. Graph the first inequality

Draw a solid line $27x+16y=80$ and the point $(0,0)$ does not satisfy the inequality $27x+16y\ge 80$. So shade the region black that does not contain the point $(0,0)$ to get the graph.

Part (b) Step 2. Graph the second inequality on the same coordinate plane

Draw a solid line $3.20x+1.75y=10$ and the point $(0,0)$ satisfy the inequality $3.20x+1.75y\le 10$. So shade the region red that contains the point data-custom-editor="chemistry" $(0,0)$ to get the graph. 

The common shaded region is the solution of the system of inequality.  

Part (c) Step 1. Check the point on the graph

$3$ bottles of protein water and $1$ protein bar corresponds to the point $(3,1)$ on the coordinate plane. Plotting the point we get

The point does not lie on the common shaded region, so he cannot buy $3$ bottles of protein water and $1$ protein bar.

Part (d) Step 1. Check the point on the graph

No bottles of protein water and $5$ protein water corresponds to point $(0,5)$ on the coordinate plane. Plotting the point we get

The point lies on the common shaded region, so he can buy no bottles of protein water and $5$ protein bars.