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Linear Programming

Maximize and minimize quantities using linear inequality systems

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Compare the Popularity of Different Coffees
Teacher Contributed

How can we compare the popularity of different coffees?

Most coffee places have a lot of rotating coffee blends throughout the year. In July 2012, the most popular two types of coffee at one particular cafe were the Medium Roast Blend and the Dark Roast Blend. In one month, the Medium blend sold three times as much as the Dark blend. This determined how much the store had to order for the following month. Let’s demonstrate this relationship as a system of equations, and find the minimum amount of the Dark blend this store should order.

The store normally orders 100 pounds total of the two most popular blends every month. In this case, they’re going to order Medium and Dark blends. We also know that the amount of Medium blend shouldn't exceed more than three times the Dark blend, since that is the ratio from the prior month.

Let’s assume that \begin{align*}x\end{align*} represents the amount of the Medium blend ordered in pounds, and \begin{align*}y\end{align*} represents the amount of the Dark blend ordered in pounds.

Our inequalities are \begin{align*}x \le 3y\end{align*} and \begin{align*}x + y > 100\end{align*}

In order to graph the inequalities, we have to solve for \begin{align*}y\end{align*} in both.

We now have \begin{align*}y \ge \left(\frac{1}{3}\right) x\end{align*} and \begin{align*}y > -x + 100\end{align*}

The graph below shows the two inequalities graphed on the same set of axes.

Let’s solve this system of inequalities algebraically to find the minimum amount of the Dark blend.

For the sake of finding the lowest point of the graph, we’ll replace the inequality signs with equal signs, so that we can solve for \begin{align*}x\end{align*} and then solve for \begin{align*}y\end{align*}. Let’s use the Substitution method.

\begin{align*}\left(\frac{1}{3}\right)x &= - x + 100\\ \left(\frac{4}{3}\right)x &= 100\\ x &= 75 \ lbs\end{align*}

Now we need to solve for \begin{align*}y\end{align*}.

\begin{align*}y &= \left(\frac{1}{3}\right)(75)\\ y &= 25 \ lbs\end{align*}

The store should order a minimum of 25 lbs of the Dark blend.

Explore More

Research a local coffee shop and find out some information that you can represent as a system of linear inequalities! You will need to compare two different products, and you will need to ask a worker at the store for some of the product information.

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