## Real World Applications – Algebra I

### Topic

How can we compare the popularity of Starbucks coffee?

### Student Exploration

Starbucks has a lot of rotating coffee blends throughout the year. In July 2012, the most popular two types of coffee at one Starbucks location is the Veranda Blend and the Sumatra Blend. In one month, the Veranda blend sold three times as much as the Sumatra blend. This determined how much the store has to order for the following month. Let’s demonstrate this relationship as a system of equations, and find the minimum amount of the Sumatra 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 Veranda and Sumatra blends. We also know that the amount of Veranda blend shouldn't exceed more than three times as the Sumatra blend, since that is the ratio from the prior month.

Let’s assume that \begin{align*}x\end{align*} represents the amount of the Veranda blend ordered in pounds, and \begin{align*}y\end{align*} represents the amount of the Sumatra 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 Sumatra 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 25lbs of the Sumatra blend.

### Extension Investigation

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.