Please help with this Maximization problem As part of a weight reduction program
ID: 3194919 • Letter: P
Question
Please help with this Maximization problem
As part of a weight reduction program, a man designs a monthly exercise program consisting of bicycling, jogging, and swimming. He would like to exercise at most 28 hours, devote at most 7 hours to swimming, and jog for no more than the total number of hours bicycling and swimming. The calories burned by this person per hour by bicycling, jogging, and swimming are 200, 485, and 265, respectively. How many hours should be allotted to each activity to maximize the number of calories burned? What is the maximum number of calories he will burn? (Hint: Write the constraint involving jogging in the form so.) swimming, and jog for n moncivly. How many hours shouain in the fom o) Let x1 be the number of hours spent bicycling, let x2 be the number of hours spent jogging, and let x3 be the number of hours spent swimming. What is the objective function? To maximize the number of calories burned, the man should spend hours bicycling, hours jogging, and hours swimming. (Simplify your answers.) He will burn a maximum of calories (Simplify your answer.)Explanation / Answer
Let the number of bicycling hours be x1
Let the number of jogging hours be x2
Let the number of swimming hours be x3
Maximum Calories z = 200x1 + 485x2 + 265x3
Constraints:
x1 + x2 + x3 <= 28
x3 <= 7
x2 <= x1 + x3
x2 - (x1 + x3) <= 0 [constraint for jogging less than zero]
Now the maximum amount of calories burned is in jogging, hence we need to maximize the value of jogging
since max(x2) = x1 + x3
so we can write
x1 + x2 + x3 <= 28
x2 + x2 = 28, x2 = 14 hours
Now since swimming has higher calories than bicycling, hence we maximizing swimming but it is bounded by x3=7
Hence x1 + 14 + 7 = 28
x1 = 7 hours
The man should spend 7 hours bicycling, 14 hours jogging and 7 hours swimming
Total calories burned = 200(7) + 485(14) + 265(7) = 10045 calories
He will burn a maximum of 10045 calories