As part of a weight reduction program, a man designs a monthly exercise program

Question

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 26 hours, devote at most 9 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, 562, and 278, 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 less than or equals 0.) Let x 1 be the number of hours spent bicycling, let x 2 be the number of hours spent jogging, and let x 3 be the number of hours spent swimming. What is the objective function?

Z=

To maximize the number of calories burned, the man should spend ___hours bicycling, ____ hours jogging, and ____ hours swimming.

Explanation / Answer

Solution:

B = hours biking
J = hours jogging
S = hours swimming

R = biking rate = 200 cal/h
Rj = jogging rate = 562 cal/h
Rs = swimming rate = 278 cal/h

B + J + S 26

B + J + S = 26 ... since: most hours = most calories bur ned

We would like to maximize J, since Rj is the highest rate however there are
other constraints:

J B + S and S 9

To maximize J, we must maximize B + S

Since: S 9 and Rs > R then S must be maximized ...
S = 9 hours
... because swimming bur ns more calories compared to biking.

J B + 9
J – B 9 ... since: Rj > R then (J – B) must be maximized
J – B = 9
J = B + 9

B + S + J = 26
B + 9 + (B + 9) = 26
2B + 18 = 26
B = 4 hours
Now,

J = B + 9
J = 4 + 9
J = 13 hours

Let B = X1 , J = X2 and S= X3

Objective function Z = 200X1 + 562X2 + 278X3

To maximize the number of calories burned, the man should spend 4 hours bicycling, 13 hours jogging, and 9 hours swimming.

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