I need Algebra 2 help...?

Mr. B. Ball, director of manufacturing for the High Step Shoe Corporation wants to maximize the company’s profits. The company makes two brands of sports shoe, Airheads and Groundeds. The company earns $10 profit on each pair of Airheads and $8.50 profit on each pair of Groundeds. The company must cut the materials on a machine and have workers assemble the pieces into shoes. There are 6 machines to cut the materials, 850 workers that assemble the shoes, and the assembly plant works a 40 hour week. Each hour, each cutting machine can do 50 minutes of work. Each pair of Airheads requires 3 minutes of cutting time and 7 hours of assembly time. Each pair of Groundeds requires 2 minutes of cutting time and 8 hours of assembly time.





How many of each type of shoe should the company produce to maximize its profits?





This is an example of a linear programming problem. Believe it or not, you already know most of the mathematics required to solve this problem! To see how it is done, go to this link and complete the High Step Shoes Activity:

I need Algebra 2 help...?
Let's call A=# of Airheads and G=# of Groundeds.





We want to maximize profit, meaning


Maximize P=10A+8.5G





The total amount of available machine time is


6 machines x 50 minutes/machine/hour x 40 hours/week = 12000 minutes/week





The total amount of available worker assembly time is


850 workers x 40 hours/week/worker = 34000 hours/week





We can write 2 statements for the shoes:


Machine minutes required for all the shoes %26lt; 12000


3A+2G%26lt;12000


Assembly hours required for all the shoes %26lt; 34000


7A+8G%26lt;34000





We can represent these equations on a graph (with A on one axis and G on the other) and the area under the corresponding line is where A and G are valid. Also remember that you can't make negative shoes so we only need to focus on the quadrant where both A and G are positive. The part of the quadrant that is under the lines of both equations are the only solutions that satisfy both equations.





If you do the plotting correctly, you'll have a 4 sided shape whose corners are (A,G)=(0,0), (4000,0), (0,4250), and (?,?).


The last point is the intersection of the two equations.





Obviously, to maximize profits, you want to make as many shoes as you possibly can, so we can rule out (0,0) as a plausible answer. Therefore, you're left with 3 combinations of (A,G) to try. Plug them into the profit equation and the one that gives you the largest answer is the combination to maximize profits.



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