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Chapter 11: Q. 26 (page 791)

Ice Cream The Mom and Pop Ice Cream Company makes two kinds of chocolate ice cream: regular and premium. The properties of 1 gallon (gal) of each type are shown in the table:

In addition, current commitments require the company to make at least 1 gal of premium for every 4 gal of regular. Each day, the company has available 725 pounds (lb) of flavoring and 425lbof milk-fat products. If the company can ship no more than 3000 lb of product per day, how many gallons of each type should be produced daily to maximize profit?

Short Answer

Expert verified

The Mom and Pop Company gets maximum profit when the company makes no regular ice cream at all and 340 gallons of premium ice cream. The maximum profit obtained is $306.

Step by step solution

01

Step 1. Given information

Let number of gallons of regular and premium ice-cream produced by Mom and Pop Ice cream Company bexandy.
The regular flavour requires 24ozof flavouring contents and 12ozof milk-fat products while the premium flavour requires 20ozof flavouring contents and 20ozof milk-fat products. The company has a total 725 pounds of flavouring contents and 425 pounds of milk-fat products.

1 pound=16 ounces

24x+12y鈮�725脳1624x+12y鈮�1160020x+20y鈮�425脳1620x+20y鈮�6800

02

Step 2. Finding the value of P

The shipping weight of regular flavour is 5 pounds and that of premium flavour is 6 pounds. The company can ship no more than 3000 pounds per day.
5x+6y鈮�3000
The company is required to make at least 1 gallon of premium flavour for every 4 gallon of regular flavoured ice-cream.
4y鈮�x
The profit over the regular ice-cream per gallon per day isand that over the premium flavour is. We can obtain a profit function that can be maximised in order to solve the given linear programming problem.
P=0.75x+0.90y
03

Step 3. Sketch the graph of the inequalities obtained in rectangular Cartesian coordinate system. 

Now,evaluate the profit function at each of the corners of the feasible region obtained in the above graph.
Vertex Value of profit function
P=0.75x+0.90y
(0,0) P=0.75(0)+0.90(0)=0
(272,68) P=0.75(272)+0.90(68)=265.2
(0,340) P=0.75(0)+0.90(340)=306

The Mom and Pop Company gets maximum profit when the company makes no regular ice cream at all and 340 gallons of premium ice cream. The maximum profit obtained is$306

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