Question

A fruit juice company makes two kinds of juice blends, each in 1 gallon bottles. the regular mix uses 1/2 a 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 mix made using the 225 gallons of orange juice and 150 gallons of pineapple juice it currently has on hand

A. Give four constraints 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 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. The maximum profit is the maximum value found when evaluating the profit function at each vertex on the graph ( or the point closest to the vertex with values that make sense in context.) Whats 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

56 regular 37 tropical Maximum profit $204.75

Explanation:

Constraints are:

r ≥ 0

t ≥ 0

2r+ 3t ≤ 125

2r + 1t ≤ 150

I will multiply the second by -3 to cancel t variable and see what is the maximum amount for the regular

2r + 3t ≤ 225

-6r - 3t ≤ -450

ADD BOTH

-4r ≤ -225

r≤ 56.5

Since we need to pick whole bottles ,not a fraction of a bottle we need ro round down.It should be less than 65.5.So we need 56 bottles of regular

2*56 +3t ≤ 225

112+3t ≤ 225

3t ≤ 113

t≤ 37.6666

37 bottles of tropical

2*56 + 3*37 = 112+111 =223 223<225

2*56 +37 = 112 +37 =149 149<150

$2.5*56 = $140

$1.75*37 = $64.65

Total profit $204.75