Question

A fruit juice company makes two kinds of juice blends, each in 1-gallon bottles. the regular mixes 1/2 gallon of orange juice and 1/2 gallon of pineapple juice, while the tropical mix uses 3/4 gallon of orange juice and 1/4 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 225 gallons of orange juice and 150 gallons of pineapple juice it currently has on hand.

a. Give four constrants the company must take into consideration to maximize the profit.

b. Graph and shade the region bounded by the constraints from part a.

c. If the regular juice blend makes a $2.50 profit per bottle sold and the tropical juice 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. the maximum profit maximum value found when evaluating the profit function at each vertex on the graph (or the closest to the vertex with values that makes sense in context). 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

a) In making bottles of regular he needs of orange and of pine apple. In making bottles of tropical he needs of orange and of pineapple. All in gallons.

Thus, he uses of orange and of pineapple; also, note that both being quantities have to be non-negative. His constraints are

b) The graph is here:

The region bounded by the red boundary in the graph is the region of the constraints. The line pointed by orange arrow is the line

The line pointed by green arrow is the line

The horizontal axis is -axis.

c) The profit function is . At the vertex its value is ; at the vertex its value is . Finally, the vertex of intersection of the above two lines is given by (solve these equations)

and here the value of is . The other vertex is the origin where value of is zero. Thus, the maximum profit is which occurs when , meaning he needs to sell bottles of regular and of tropical.