Q. 26 - Precalculus Enhanced with Graphing Utilities

Question

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 $425lb$ of 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? 

Step-by-Step Solution

Answer

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 1. Given information

Let number of gallons of regular and premium ice-cream produced by Mom and Pop Ice cream Company be x and y.

The regular flavour requires $24oz$ of flavouring contents and $12oz$ of milk-fat products while the premium flavour requires $20oz$ of flavouring contents and $20oz$ of milk-fat products. The company has a total 725 pounds of flavouring contents and 425 pounds of milk-fat products.

1 pound=16 ounces

$$\begin{gathered} 24x+12y\le 725\times 16 \\ 24x+12y\le 11600 \\ 20x+20y\le 425\times 16 \\ 20x+20y\le 6800 \end{gathered}$$

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.

$5x+6y\le 3000$

The company is required to make at least 1 gallon of premium flavour for every 4 gallon of regular flavoured ice-cream.

$4y\ge x$

The profit over the regular ice-cream per gallon per day is  and 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.

$P=0.75x+0.90y$

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 $P=0.75x+0.90y$
$(0,0)$	$P=0.75\left(0\right)+0.90\left(0\right)=0$
$(272,68)$	$P=0.75\left(272\right)+0.90\left(68\right)=265.2$
$(0,340)$	$P=0.75\left(0\right)+0.90\left(340\right)=306$

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 $\$306$