# Math Help - This Wicked Coffee linear equation.

1. ## This Wicked Coffee linear equation.

This is my first and likely only post on a forum like this: I was asked to help someone with this Math problem and normally I can answer a question here and there to get a student by, but this question from her online practice site really got me....I haven't done this in a long time--would someone be kind enough to walk me through it, so I in turn can walk her through it? Math is not my occupation, so I appreciate the history lesson!! (I'm ok with not getting an answer, if someone can show me what type of a problem to set up) Thank You so much.

For the current month, Sue's Coffee House has available 89 pounds of A grade coffee and 115 pounds of B grade coffee. These will be blended into 1 -pound packages as follows: an economy blend that contains 6 ounces of A grade coffee and 10 ounces of B grade coffee, and a superior blend that contains 8 ounces of A grade coffee and 8 ounces of B grade coffee. There is a .50 profit on each economy blend package sold and a .75 profit on each superior blend package sold. Assuming that the store is able to sell as many blends as they make, how many packages of each blend should they make in order to maximize their profit for the month?

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For the current month, Sue's Coffee House has available 89 pounds of A grade coffee and 115 pounds of B grade coffee. These will be blended into 1 -pound packages as follows: an economy blend that contains 6 ounces of A grade coffee and 10 ounces of B grade coffee, and a superior blend that contains 8 ounces of A grade coffee and 8 ounces of B grade coffee. There is a .50 profit on each economy blend package sold and a .75 profit on each superior blend package sold. Assuming that the store is able to sell as many blends as they make, how many packages of each blend should they make in order to maximize their profit for the month?
1.
x = number of packages of eco blend
y = number of packages of sup blend

2.
Used amount of grade A: $\dfrac6{16} x + \dfrac{10}{16} y \leq 89~\implies~y\leq -\dfrac35 x + \dfrac{712}5$ black line
Used amount of grade B: $\dfrac8{16} x + \dfrac{8}{16} y \leq 115~\implies~ y\leq -x+230$ green line

Since each new package contains 1 lb you have to keep in account:

$x+y\leq 89 + 115 ~\implies~y\leq -x+204$ pink line

3. The profit function is:

$P = 0.5x + 0.75y ~\implies~y = -\dfrac23 x+ \dfrac43 P$ red line

4. The last equation describes a family of straight lines. You are looking for that line which has at least one point in common with the "planning polygon" (this is the literal translation from German for this area) and whose y-intercept is the largest. The optimal point of the "planning polygon" is Q(154, 50). This yields the red line. The y-intercept is :

$50 = -\dfrac23 \cdot 154 +\dfrac43 P~\implies \dfrac43 P = \dfrac{458}3~\implies~\boxed{P = \114.50}$