Question 1175143: Background info: Imagine you are trying to maximize the calories you burn in a 60-minute workout you do a few times a week. Running burns 9 calories per minute, aerobics burns 6 calories per minute, and rowing burns 7 calories per minute. You want to perform all three exercises to work different muscle groups. For the best effect, you need to run for at least 5 minutes and row for at least 15 minutes. Your aerobics session should be no more than 30 minutes. How many minutes should you perform each exercise to burn the maximum calories?
Question: Convert the inequalities into equations, and then use the substitution method to find four possible vertices in the form (x, y, z).
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website! I see three posts, presenting the same problem slightly differently each time. I will guess these are from the same reader....
I will ignore all but the first paragraph of this post, since the language is not clear as to what you are supposed to do. I will simply address the problem posed in the first paragraph.
Let x be the number of minutes running
let y be the number of minutes doing aerobics
let z be the number of minutes rowing
The given constraints are...
x >= 5
y <= 30
z >= 15
x+y+z = 60
The objective function is the number of calories you burn, 9 per minute running, 6 per minute doing aerobics, and 7 per minute rowing: 9x+6y+7z
This problem doesn't require any formal mathematical methods.
Doing aerobics burns the fewest calories per minute, so to maximize the number of calories burned you should do aerobics as little as possible. Since the constraint on doing aerobics is only that they should be done for no MORE than 30 minutes, the maximum number of calories will be burned if aerobics are done for 0 minutes.
That leaves all 60 minutes for running and rowing.
Then running burns more calories per minute than rowing, so as much time as possible should be used running. The constraints require you to row for at least 15 minutes; so to burn the maximum number of calories you should row for EXACTLY 15 minutes and run for the other 45 minutes.
ANSWER: Burn the maximum number of calories under the given constraints by rowing for 15 minutes and running for 45 minutes. That maximum number of calories burned is 15(7)+45(9) = 105+405 = 510.
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