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Maria 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 milk has 7 grams of protein and 110 calories. An ounce of cheese has 11 grams of protein and 22 calories. Let M be the number of ounces of milk, and C be the number of ounces of cheese. (2 pts) A. Write a system of linear inequalities in two variables that expresses the relationship between the two types of food. B. Solve the system in part (A). Make sure you label the vertices with exact coordinates.

Answer :

A. The system of linear inequalities in two variables that expresses the relationship between the two types of food is [tex]7M + 11C & \geq 35; 110M + 22C & \leq 200, M & \geq 0 \text {and} C & \geq 0 \\[/tex]

B. The value of C is [tex]\[\frac{35 - 7M}{11} \leq C \leq \frac{200 - 110M}{22}\][/tex]

and [tex]\[\frac{35 - 7M}{11} = \frac{200 - 110M}{22}\].[/tex]

A. The problem can be solved using linear inequalities based on the given information. Let's define the variables:

- M = Number of ounces of milk

- C= Number of ounces of cheese

The objective is to increase protein consumption by at least 35 grams[tex](\(7M + 11C \geq 35\))[/tex] and increase caloric intake by no more than 200 calories [tex](\(110M + 22C \leq 200\)).[/tex]

So, the system of linear inequalities is as follows:

[tex]7M + 11C & \geq 35 \\110M + 22C & \leq 200 \\M & \geq 0 \\C & \geq 0 \\[/tex]

B. Now, let's solve the system of inequalities to find the feasible region and its vertices.

1. Solve the first inequality for C:

[tex]\[C \geq \frac{35 - 7M}{11}\][/tex]

2. Solve the second inequality for C:

[tex]\[C \leq \frac{200 - 110M}{22}\][/tex]

3. Plot the lines corresponding to [tex]\(C = \frac{35 - 7M}{11}\) and \(C = \frac{200 - 110M}{22}\)[/tex]

4. The feasible region is the overlapping region between the two inequalities:

[tex]\[\frac{35 - 7M}{11} \leq C \leq \frac{200 - 110M}{22}\][/tex]

5. Find the values of M at the intersection points of the lines:

[tex]\[\frac{35 - 7M}{11} = \frac{200 - 110M}{22}\][/tex]

Remember that M and C cannot be negative [tex](\(M \geq 0\)[/tex] and [tex]\(C \geq 0\)),[/tex] so you only need to consider the feasible region in the first quadrant of the coordinate plane.

Learn more about linear inequalities here:

https://brainly.com/question/23093488

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