SOLUTION: A manufacturer produces two models of mountain bikes. Model A requires 5 hours of assembly time and 2 hours of painting time, and Model B requires 4 hours of assembly time and 3 ho

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Question 1198872: A manufacturer produces two models of mountain bikes. Model A requires 5 hours of assembly time and 2 hours of painting time, and Model B requires 4 hours of assembly time and 3 hours of painting time. The maximum total weekly hours available in the assembly department and the painting department are 200 hours and 108 hours, respectively. The profits per unit are $25 for Model A and $15 for Model B. How many of each type should be produced to maximize profit?
Bikes of Model A=
Bikes of Model B=
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
let x = number of model A bikes.
let y = number of model B bikes.

objective function is:
profit = 25x + 15y

constraint inequalities are:
5x + 4y <= 200
2x + 3y <= 108


it helps to make a chart such as the one below: 

                       model A           model B
number of bikes           x                 y
assembly time             5                 4            <= 200
painting time             2                 3            <= 108


this way it's easier to see what hours belong where.

additional constraints are that x and y must both be greater than or equal to 0.

using the desmos.com calculator, you would graph the opposites of the inequalities.

the area of the graph that is not shaded would be the region of feasibility.

you would evaluate the objective function at the corner points of this region.

the maximum profit will be at one of those corner points.

after evaluating the objective function at each corner pooint, you will find that the maximum profit is at (40,0).

this means 40 model A mountain bikes and no model B mountain bikes should be made to maximize profit, assuming all mountain bikes are sold.

the constraint inequalities must also be satisfied at each corner point.
at (40,0), .....

assembly time would be equal to 40 * 5 = 200
painting time would be equal to 40 * 2 = 80

both of these are within limits.

this is what the graph looks like.



the desmos.com graphing software can be found at https://www.desmos.com/calculator