Linear Programming

Question

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?

1. Write a system of inequalities based on the given constraints. Then, note the objective function.
2. Convert the inequalities into equations, and then use the substitution method to find four possible vertices in the form (x, y, z).
3. Test the four possible vertices that you found in the objective function in part A. Use those values to determine which set of values maximizes the objective function. Show your work and final answer in the space provided.

Answer 1

The function to be maximize is:

Calories Burned = (Run Calories)(Run Minutes) + (Aerobic Calories)(Aerobic Minutes)

+ (Row Calories)(Row minutes)

Run Minutes >= 5 Minutes

Row Minutes >= 15 Minutes

Row Minutes + Aerobic Minutes + Run Minutes <= 30

When solved with EXCEL Solver, Max Calories = 240

Row Minutes = 15

Run Minutes = 15

Aerobic Minutes = 0

Answer 2

Objective function C(x,y,z) = 9x + 6y + 7z

x>=5, z>=15, y<=30

x+y+z <= 60

(15,30,15) ---> 135+180+105 = 420

(45,0,15) ---> 495 + 105 = 600

(5,30,25) --> 45 + 180 + 175 = 400

(5,15,40) ---> 45 + 90 + 280 = 415