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A florist has to order roses and carnations for Valentine's Day. The florist needs to decide how many of dozens of roses and carnations should be ordered to obtain a maximum profit.
By graphing the constraints, we get the corner points of the feasible region: (10,50), (17.5,20), (0,60) and (0,20).
Now, plug these coordinates into the equation for profit.
1. (10,50)
P = 20x + 8y
P = 20(10) + 8(50)
P = 200 + 400
P = 600
This information shows us that if the florist were to order 10 dozens of roses and 50 dozens of carnations, she would make a profit of $600.
First, we assign our variables and find the objective quantity.
Let x = the number of dozens of roses purchased
Let y = the number of dozens of carnations purchased
Let P = profit
2. (17.5,20); assuming you cannot buy a half a dozen of roses we will round 17.5 to 17.
P = 20x + 8y
P = 20(17) + 8(20)
P = 340 + 160
P = 500
This information shows us that if the florist were to order 17 dozens of roses and 20 dozens of carnations, she would make a profit of $500.
By using linear programming, I was able to discover that the florist can order 10 dozens of roses and 50 dozens of carnations to achieve a maximum profit of $600.
Since the problem asks us to obtain the maximum profit, the objective quantity is: maximum profit.
Next, we express the constraints as inequalities and graph them to find the corner points of the feasible region.
3. (0,20)
P = 20x + 8y
P = 20(0) + 8(20)
P = 0 + 160
P = 160
This information shows us that if the florist were to order 0 dozens of roses and 20 dozens of carnations, she would make a profit of $160.
2. (0,60)
P = 20x + 8y
P = 20(0) + 8(60)
P = 0 + 480
P = 480
This information shows us that if the florist were to order 0 dozens of roses and 60 dozens of carnations, she would make a profit of $480.
Constraints:
Let's also take note that the equation for the florist's profit is:
P = 20x + 8y
We know this because the profit of the number of dozens of roses sold is 20x. This is added to the overall profit of dozens of carnations sold, which can be represented as 8y. Together they make the overall profit.