I have just one more linear programming question, and then I think I'll be good. Thanks to everyone who answered my previous question, I am beginning to understand now.
Suppose you make and sell skin lotion. A quart of regualr skin lotion contains 2cups of oil, and one cup of cocoa butter. A quart of extra rich skin lotion contains 1 cup oil, and 2 cups of cocoa butter. You will make a profit of 10 per quart on regualar lotion, and a profit of $8 a quart on extra rich lotion. You have 24 cups oil, and 18 cups cocoa butter.
1. How many quarts of each type of lotion should you make to maximize your profit?
2. What is the maximum profit?
Thanks again
r = regular (y axis)
e = extra rich (x axis)
p = profit = 8 e + 10 r
oil limit line (24 max)
oil = 2 r + 1 e, when e = 0, r limit is 12
when r = 0, e = 24
so oil line is
r = 12 - .5 e
butter limit line (18 max)
butter = 1 r + 2 e, when e = 0, r limit is 18
when r = 0, e limit is 9
so butter line is
r = 18 - 2 e
where is intersection of oil limit and butter limit lines?
4 r = 48 - 2 e
1 r = 18 = 2 e
-------------
3 r = 30
r =10
e = 4 so at (4,10)
so points
(0,0)
(0,12)
(4,10)
(9,0)
calculate profit = 8 e + 10 r at each point
32 + 100 = 132 at (4,10) wins
Thank you Damon, your answer for this problem, and the other one was very thorough and easy to understand.
To solve this linear programming problem, we need to set up a system of inequalities based on the given constraints and objective function.
Let's define the decision variables:
Let x be the number of quarts of regular lotion to make.
Let y be the number of quarts of extra rich lotion to make.
Objective function:
We want to maximize the profit, so our objective function can be defined as:
Profit = 10x + 8y
Constraints:
1. The amount of oil used in regular lotion: 2 cups per quart. So, the total cups of oil used must be less than or equal to the available oil, which is 24 cups. The inequality for this constraint is:
2x + y ≤ 24
2. The amount of cocoa butter used in regular lotion: 1 cup per quart. So, the total cups of cocoa butter used must be less than or equal to the available cocoa butter, which is 18 cups:
x + 2y ≤ 18
3. We cannot produce negative quantities of lotion: x ≥ 0 and y ≥ 0
Now, let's solve this system of inequalities to find the optimal solution.
First, we need to plot the feasible region on a graph.
1. Draw the line 2x + y = 24 (to find the feasible region for oil constraint).
2. Draw the line x + 2y = 18 (to find the feasible region for cocoa butter constraint).
Both lines will intersect in the first quadrant, and the region bounded by these lines and the axes will be our feasible region.
Next, we need to evaluate the objective function at the vertices of the feasible region to determine the maximum profit.
1. Vertices (0, 0):
Profit = 10(0) + 8(0) = 0
2. Vertices (0, 9):
Profit = 10(0) + 8(9) = 72
3. Vertices (12, 0):
Profit = 10(12) + 8(0) = 120
The maximum profit is obtained at the vertex (12, 0) with a profit of $120.
Therefore, to maximize profit, you should make 12 quarts of regular lotion and 0 quarts of extra rich lotion.