Question 318404: Machine A can produce 30 steering wheels per hour at a cost of $8 per hour. Machine B can produce 40 steering wheels per hour at a cost of $12 per hour.
The company can use either machine by itself or both machines at the same time. What is the minimum number of hours needed to produce 380 steering wheels if the cost must be no more than $108?
Answer by Fombitz(32388)   (Show Source):
You can put this solution on YOUR website! Let X be the number of hours using Machine A.
Let Y be the number of hours using Machine B.
Total quantity:
Total costs:
You must satisfy both equations : making the number of steering wheels and keeping cost below or equal to a certain limit.
Find the point of intersection between the two curves,
1.
2.
Multiply eq. 2 by (-3) and add to eq. 1,
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Then use either equation to solve for Y,
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Run machine A for 6 hours, machine B for 5 hours to make 380 steering wheels at a total cost of $108.
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Additionally, this is actually a linear programming problem.
We should better define the feasible region, although the region is not a region but a line segment.
The red line shows the cost limit while the green line shows the number of steering wheels.
Once the green line crosses the red line, the total cost inequality does not hold.
So the only points we need to check are the intersection point between the red and green curve and the intersection point between the green line and the x-axis.
What we're trying to minimize is the number of hours total, 
At  ,  , 
When  ,  ,  and then 
So the minimum for does occur at ,
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