SOLUTION: Please help a manufacturer produces bikes and wagons. the time (in hours) required for assembling and painting is shown in the following table: process hours(bike)

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Question 536571: Please help
a manufacturer produces bikes and wagons. the time (in hours) required for assembling and painting is shown in the following table:
process hours(bike) hours(wagon)
assembling 3 4
painting 5 7
the total time available for assembling and painting is 400 hours and 300 hours, respectively. the profit per unit is $20 for a bike and $34 for a wagon. find the number of bikes and wagons that should be produced in order to yield maximum profit. let x be the number of bikes produce and Y be the number of wagons produced...Formulate the linear programming problem
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
The time required for assembling x bikes and y wagons is 3x%2B4y
The time required for painting x bikes and y wagons is 5x%2B7y
The profit from x bikes and y wagons is P=20x%2B34y
The constrains are x%3E=0 y%3E=0 3x%2B4y%3C=400 5x%2B7y%3C=300
The first two tell you that you will not produce a negative number of bikes or a negative number of wagons, so your feasibility region excludes anything to the left of the y-axis and/or below the x-axis. The other two boundaries are lines that cross the positive x- and y-axis. If those two lines may intersect at a point with positive x and y, that point will be a corner of the feasibility region.
graph%28+400%2C+400%2C+-50%2C+150%2C+-50%2C+150%2C+100-3x%2F4%2C+300%2F7-5x%2F7+%29 The green line represents 5x%2B7y=300 (painting time constraint), the blue one 3x%2B4y=400 (assembling constraint)
Unfortunately, your feasibility region is the triangle between the green line and the axes. It seems to be that no matter how many items you can assemble, the painting is your bottleneck. If possible, expand the painting section of the shop and hire more painters.