How many quarts of each flavor should you make to earn the largest profit?

Question

You manage an ice cream factory that makes two flavors: Creamy Vanilla and Continental Mocha. Into each quart of Creamy Vanilla go 2 eggs and 3 cups of cream. Into each quart of Continental Mocha go 1 egg and 3 cups of cream. You have in stock 550 eggs and 900 cups of cream. You make a profit of $3 on each quart of Creamy Vanilla and $2 on each quart of Continental Mocha. How many quarts of each flavor should you make to earn the largest profit? HINT [See Example 2.] (If an answer does not exist, enter DNE.)

Creamy Vanilla quarts =?

Continental Mocha quarts =?

Answer 1

It is easy to imagine that we need to use up either all the eggs, or all the cream (or both). If we think that we've reached a maximum profit and there is both some cream and some eggs left, then we can still make a non-zero quantity of ice cream (of either flavor) and make more profit.

Let's call v the number of quarts of vanilla ice cream we make, and m the number of quarts of mocha ice cream. Let's also call p the profit.

Let's think in terms of eggs first. We know we only have 550, and it takes 2 to make one quart of vanilla, and one to make one quart of mocha. That gives us an inequality:

2v + m <= 550

Similarly, for cream, we get:

3v + 3m <= 900 which we can simplify into v + m <= 300

We also know the profit we will make is such that:

p = 3v + 2m

Let's first assume that we use up all the cream. We'll see where that leads us, and if there are some leftover eggs, we'll try that other route and see if it leads to a higher profit.

So if we use up all the cream, one of the inequalities above becomes an equality:

v + m = 300 or v = 300 - m

Let's replace that value of v into the other inequality:

2 * (300 - m) + m <= 550

After a couple of lines of unwrapping that, that leads us to m >= 50

And we can also express the profit in terms of m:

p = 900 - m

So if we want to maximize the profit, we need to minimize m, but we can't let it go lower than 50. So we pick m = 50, which leads to v = 300 - 50 = 250, and p = 850. At this point, we can see that we have used up both all the cream and all the eggs, so we wouldn't get any more profit starting with the other hypothesis of using up all the eggs.

So the final answer is:

Make 250 quarts of creamy vanilla, and make 50 quarts of continental mocha. That will lead to a $850 profit.