Question 1206214: The Wellbuilt Company produces two types of wood chippers, economy and deluxe. The deluxe model requires 3 hours to assemble and
1
2
hour to paint, and the economy model requires 2 hours to assemble and 1 hour to paint. The maximum number of assembly hours available is 24 per day, and the maximum number of painting hours available is 8 per day. If the profit on the deluxe model is $96 per unit and the profit on the economy model is $74 per unit, how many units of each model will maximize profit?
______ deluxe units
______ economy units
Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! my interpretation of your problem statement is:
The Wellbuilt Company produces two types of wood chippers, economy and deluxe.
The deluxe model requires 3 hours to assemble and (i think) 2 hours to paint, and the economy model requires 2 hours to assemble and 1 hour to paint.
The maximum number of assembly hours available is 24 per day, and the maximum number of painting hours available is 8 per day. If the profit on the deluxe model is $96 per unit and the profit on the economy model is $74 per unit, how many units of each model will maximize profit?
the first interpretation table is shown below.
economy model deluxe model
assemble hours 2 3 <= 24
paint hours 1 2 <= 8
profit dollars 74 96 maximize
the first interpretation constraint inequalities are:
2x + 3y <= 24
x + 2y <= 8
the second interpretation would be 1 hours to paint for the deluxe model rather than 2. it wasn't clear which one you wanted, so i did both and you can decide which of the two was what you really wanted, if either.
the second interpretation table is shown below.
economy model deluxe model
assemble hours 2 3 <= 24
paint hours 1 1 <= 8
profit dollars 74 96 maximize
the second interpretation constraint inequalities are:
2x + 3y <= 24
x + y <= 8
using the desmos.com calculator (https://www.desmos.com/calculator), your would graph the opposite of the inequalities.
the area on the graph that is not shaded is your region of feasibility.
the corner points of this area is where your maximum profit will be.
the first interpretation graph is shown below.
the second interpreation graph is shown blow.
the first interpretation says that 8 economy wood chippers give more profit.
(0,4) = 384 profit
(8,0) = 592 profit
the second interpretation says that 8 deluxe wppd chippers give more profit.
(0,8) = 768 profit
(8,0) = 592 profit
the difference is in the number of hours to paint the deluxe model.
if it's 2 hours, then the economy chipper gives more profit.
if it's 1 hour, then the deluxe chipper gives more profit.
the painting hours constraints make that happen.
total painting hours are less than or equal to 8.
if it takes 2 hours to paint one deluxe model, then only 4 can be done.
if it takes 1 hour to paint one deluxe model, then 8 can be done.
since profit on the deluxe model is greater than profit on the economy model, being able to make more deluxe models generates more profit overall.
bottom line on this problem.
if you assume 2 hours to paint the deluxe model, then maximum profit would be when you make and sell 8 economy models for a total profit of 592 dollars.
if you assume 1 hour to paint the deluxe model, then maximum profit would be when you make and sell 8 deluxe models for a total profit of 768 dollars.
if you, in fact, wanted to analyze both 1 hour and 2 hours to paint the deluxe model, you got it.
let me know if you have any questions or need further clarification.
theo
Answer by ikleyn(52786)  (Show Source):
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