Q 324 - Intermediate Algebra

Question

Jocelyn desires to increase both her protein consumption and caloric intake. She desires to have at least $35$ more grams of protein each day and no more than an additional $200$ calories daily. An ounce of cheddar cheese has $7$ grams of protein and $110$ calories. An ounce of parmesan cheese has $11$ grams of protein and $22$ calories.

ⓐ Write a system of inequalities to model this situation. 

ⓑ Graph the system. 

ⓒ Could she eat $1$ ounce of cheddar cheese and $3$ ounces of parmesan cheese?

ⓓ Could she eat $2$ ounces of cheddar cheese and $1$ ounce of parmesan cheese? 

Step-by-Step Solution

Answer

Part a. A system of inequalities to model the situation is $\left\{\begin{array}{l}7x+11y\ge 35\\ 110x+22y\le 200\\ x\ge 0\\ y\ge 0\end{array}\right.$

Part b. The graph of the system is

Part c. Yes, she can eat $1$ ounce of cheddar cheese and $3$ ounces of parmesan cheese.

Part d. No, she cannot eat $2$ ounces of cheddar cheese and $1$ ounce of parmesan cheese

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

Let $x$ represents the number of ounces of cheddar cheese bought and $y$ represents the number of ounces of parmesan cheese bought.

One ounce of cheddar cheese contains $7$ grams of protein and one ounce of parmesan cheese contains $11$ grams of protein. She needs to intake at least $35$ grams of protein, so an inequality can be written as

$7x+11y\ge 35$

One ounce of cheddar cheese contains $110$ calories and one ounce of parmesan cheese contains $22$ calories. She needs to intake no more than $200$ calories, so an inequality can be written as

$110x+22y\le 200$

Now number of ounces will always be a non-negative quantity, so the system of inequalities is  

$\left\{\begin{array}{l}7x+11y\ge 35\\ 110x+22y\le 200\\ x\ge 0\\ y\ge 0\end{array}\right.$

Part (b) Step 1. Graph the first inequality

Draw a solid line $7x+11y=35$ and the point $(0,0)$ does not satisfy the inequality $7x+11y\ge 35$. 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 $110x+22y=200$ and the point $(0,0)$ satisfy the inequality $110x+22y\le 200$. So shade the region red that contains the point $(0,0)$ 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

$1$ ounce of cheddar cheese and $3$ ounces of parmesan cheese represents the point $(1,3)$ on the coordinate plane. On plotting the point we get

The point lies in the common shaded region, so she can eat one ounce of cheddar cheese and three-ounce of parmesan cheese.

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

$2$ ounces of cheddar cheese and $1$ ounce of parmesan cheese represents the point $(2,1)$ on the coordinate plane. On plotting the point we get 

The point does not lie on the common shaded region, so she cannot eat two ounces of cheddar cheese and one ounce of parmesan cheese.