Linear inequalities

Question

A fruit juice company makes two kinds of juice blends, each in 1-gallon bottles. The regular mix uses ½ gallon of orange juice and ½ gallon of pineapple juice, while the tropical mix uses ¾ gallon of orange juice and ¼ gallon of pineapple juice. The company wants to maximize its profit from selling r bottles of the regular juice mix and t bottles of the tropical juice mix made using the 225 gallons of orange juice and 150 gallons of pineapple juice it currently has on hand. (A-CED.A.2 and A-CED.A.3)

1. Give four constraints the company must take into consideration to maximize the profit.
2. If the regular juice blend makes a $2.50 profit per bottle sold and the tropical juice blend makes a $1.75 profit per bottle sold, the profit from selling r bottles of the regular blend and t bottles of the tropical blend is P = 2.5r + 1.75t. What's the maximum profit, and how many bottles of each type of juice blend will be sold to make that profit? Assume all manufactured bottles are sold.

Answer 1

Hi Devin!

If I was the head of this juice company, I'd figure on making R gallons of regular mix, and T gallons of tropical mix. I'd draw out R and T on a two-dimensional grid, because the constraints will make a space upon that grid.

The constraints you have are as follows:

1) You cannot make a negative amount of regular mix, so R >= 0.

2) You cannot make a negative amount of tropical mix, so T >= 0.

3) You cannot use more orange juice than you have. If you make R gallons of regular mix, and T gallons of tropical mix, this will require a*R + b*T gallons of orange juice, where a and b are both constants that come from the problem. This cannot be more than OJ, the total amount of orange juice you have, so this gives a linear inequality:

aR + bT <= OJ

4) You cannot use more pineapple juice than you have. If we make an argument that is similar to our argument for Constraint #3, with c and d being coefficients and PJ being the total gallons of pineapple juice, both from the problem, this is also a linear inequality:

cR + dT <= PJ

Now, graph all four constraints on the two dimensional (R,T) grid. The last two will make a great "X" across the first quadrant of this grid, where R and T are both nonnegative. Find the point of intersection of these constraints, somewhere with R and T that are both nonzero. Follow the constraints back to the respective axes, and select the axes intercepts that make physical sense: both constraints hit either axis, but only one of the two intercepts on either axis asks for only as much pineapple or orange juice as you have; the other asks for too much and so cannot be reached. On the R axis, T is zero, so all of one or the other type of juice must go into making R. Select the amount of juice of which you have the least, and use that amount to figure out how much R you can make. Use a similar approach for T on the T axis.

You now have a quadrilateral with four points. Your objective function P will be maximized at one of these points. Evaluate it at each point, and pick your highest value.

Cheers,

Joe