Mastery Check Solutions

Show What You Know

Pups Nutrition Company is creating new dog food supplements with two main nutritional components, x and y. The Product A supplement contains 2 times as much x nutrition as y nutrition and must have at least 20 grams of nutrition per serving. The Product B supplement contains 2 times as much y nutrition as x nutrition and has at least 25 grams of nutrition per serving.

Write a system of inequalities to represent the given scenario.

$\begin{array}{rcl} Product A : 2 x + y & \geq & 20 \\ Product B : x + 2 y & \geq & 25 \\ x & \geq & 0 \\ y & \geq & 0 \end{array}$

Note

Include x ≥ 0 and y ≥ 0 because a negative value does not make sense in context.

Q: What is the minimum possible value for x and y in this problem? What inequality symbol makes sense? Explain.
A: The minimum for x and y would be 0. This means x and y will each be greater than or equal to (≥) 0.

Q: Is it possible for the values of x and y to be negative? Explain.
A: No, you cannot have a negative number of grams.

Graph your system from part A.

Note

Q: What does the shaded region represent?
A: All possible solutions that make all inequalities in the system true

Evaluate the objective function, $g (x, y) = 3 x + 5 y$ using part B.

$g (0, 20) = 3 (0) + 5 (20) = 100 g (5, 10) = 3 (5) + 5 (10) = 65 minimum g (25, 0) = 3 (25) + 5 (0) = 75$

Note

Q: Why is there no maximum value for the objective function?
A: Unbounded functions do not have a maximum.

Q: What would the region look like if the inequalities for Product A and B were less than or equal to? How would this change the graph and solution?
A: If the inequalities were less than or equal to, the solution would be a bounded region. This would mean a maximum would also occur when using the objective function.

The company wants to minimize the cost of creating each new product. Component x costs $3 per gram, while y costs $5 per gram. How much of each should they order to minimize costs?

The vertex (5, 10) is the minimum value using the objective function. Pups Nutrition Company should use 5 grams of x and 10 grams of y to minimize costs.

Say What You Know

In your own words, talk about what you have learned using the objectives for this part of the lesson and your work on this page.

Note

Restate the objectives of the lesson in your own words. If you are unable to restate the lesson objectives, go back and reread the objectives and then explain them.