Joely's Tea Shop, a store that specializes in tea blends, has available 45 pounds of A grade tea and 70 pounds of B grade tea. These will be blended into 1 pound packages as follows: A breakfast blend that contains one third of a pound of A grade tea and two thirds of a pound of B grade tea and an afternoon tea that contains one half pound of A grade tea and one half pound of B grade tea. If Joely makes a profit of $1.50 on each pound of the breakfast blend and $2.00 profit on each pound of the afternoon blend, how many pounds of each blend should she make to maximize profits? What is the maximum profit?

Question

Joely's Tea Shop, a store that specializes in tea blends, has available 45 pounds of A grade tea and 70 pounds of B grade tea. These will be blended into 1 pound packages as follows: A breakfast blend that contains one third of a pound of A grade tea and two thirds of a pound of B grade tea and an afternoon tea that contains one half pound of A grade tea and one half pound of B grade tea. If Joely makes a profit of $1.50 on each pound of the breakfast blend and $2.00 profit on each pound of the afternoon blend, how many pounds of each blend should she make to maximize profits? What is the maximum profit?

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Answer 1

To determine how many pounds of each blend Joely should make to maximize profits, we need to set up a system of inequalities.

Let's assume Joely makes x pounds of the breakfast blend and y pounds of the afternoon tea blend.

The first constraint is the availability of A grade tea. The breakfast blend requires one-third of a pound of A grade tea for each pound made, and the afternoon tea blend requires half a pound. So the first inequality is:

1/3x + 1/2y ≤ 45

The second constraint is the availability of B grade tea. The breakfast blend requires two-thirds of a pound of B grade tea for each pound made, and the afternoon tea blend also requires half a pound. So the second inequality is:

2/3x + 1/2y ≤ 70

The third constraint is that both x and y should be non-negative, as it is not possible to make a negative number of pounds of tea. So we have:

x ≥ 0 and y ≥ 0

Now, let's determine the objective function, which is the profit. The profit from the breakfast blend is $1.50 per pound, and the profit from the afternoon blend is $2.00 per pound. Thus, the objective function is:

Profit = 1.5x + 2y

Now, we have set up the linear programming problem. To solve it, we can graph the feasible region defined by the constraints and evaluate the objective function at each corner point. The point that results in the highest profit will be the solution.

However, since graphing the feasible region in this case is not efficient, we can use a more efficient method called the Simplex Method or Linear Programming software. I will now solve the problem using linear programming software.

Answer 2

This is just a standard linear programming problem. For x breakfast packages and y afternoon packages, you want to

maximize p=1.50x+2.00y subject to
x/3 + y/2 <= 45
2x/3 + y/2 <= 70

Now just use your favorite linear programming tools.