A manufacturer of skis produces two types: downhill and cross country. The times
ID: 3101213 • Letter: A
Question
A manufacturer of skis produces two types: downhill and cross country. The times required for manufacturing and finishing each ski are: manufacturing time per ski, downhill 2.5 hours, cross country 1.5 hours. Finishing time per ski: downhill 0.5 hours, cross country 1.5 hours. The maximum total weekly hours available for manufacturing and finishing the skis are 90 hours and 42 hours. The profit per ski are $50 for downhill and $50 cross country. Determine how many of each kind of ski should be produced to achieve a maximum profit?
Explanation / Answer
Since the two types are equal in value you focus on maximizing production. They both take a total of three hours to make but the down hill skis are produced unevenly. To begin set up both sets of times required in each manufacturing and finishing as a substitution, elimination, or reduction problem
2.5x + 1.5y = 90
0.5x + 1.5y = 42
Begin using reduction and subtract one equation from the other, giving you
2.5x + 1.5y = 90
-(0.5x + 1.5y = 42)= - 0.5x - 1.5y = -42
= 2.5x - 0.5x + 1.5y - 1.5y = 90 - 42
= 2.0x = 48
= x = 48 / 2 = 24 downhill skis
which gives the maximum amount of downhill skis produced in the 90 hours. Then using either substitution with x = 24 in either equation will provide the amount of cross country skis produced in both time periods.
y = 2.5(24) + 1.5y = 90
= 60 + 1.5y = 90
= 1.5y = 90 - 60
= 1.5y = 30
= y = 30 1.5
= 20 cross country skis
OR the other equation
y = 0.5(24) + 1.5y = 42
= 12 + 1.5y = 42
= 1.5y = 42 - 12
= 1.5y = 30
= y = 30/1.5
y = 20
SO 24 downhill skis is 60 hours manufacturing and 12 hours finishing where 20 cross country skis are 30 hours in manufacturing and 30 hours in finishing.
Keep in mind that it is possible to solve for y first and then x, but since they require the same amount of time elimination of cross country ( Y ) provides for x to be found easily.
Any solution set can be solved in this manner using x + y + z + .... = n
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