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?
To determine the pounds of each blend that Joely should make to maximize profits, we need to set up the problem as a linear programming problem.
Let's define the variables:
- Let x be the pounds of the breakfast blend.
- Let y be the pounds of the afternoon blend.
Now let's set up the constraints.
1) Joely has 45 pounds of A grade tea, and the breakfast blend contains one third of a pound of A grade tea. Thus, the constraint for A grade tea is:
1/3x + 1/2y <= 45
2) Joely has 70 pounds of B grade tea, and the breakfast blend contains two thirds of a pound of B grade tea. Thus, the constraint for B grade tea is:
2/3x + 1/2y <= 70
3) Both the breakfast blend and the afternoon blend need to be made in positive amounts, so the constraints for non-negativity are:
x >= 0
y >= 0
Now let's set up the objective function for maximizing profits.
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:
Maximize: 1.5x + 2y
Now we have set up the linear programming problem. We can solve it using various methods such as graphical method, simplex method, or other optimization algorithms.
Alternatively, we can use online linear programming solvers to find the optimal solution.
After solving the linear programming problem, we will obtain the values for x and y that maximize profits, as well as the maximum profit.