Respuesta :
Using objective function and linear inequalities in linear programing problem, 44 minutes of running, 16 minutes of rowing and 1 minutes of aerobics should you perform each exercise to burn the maximum of 507 calories.
According to the 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.
The system of inequality is; x + y + z = 60
x ≥ 5
y ≥ 15
0 < z ≤ 30
The objective function is, f(x, y, z) = 9·x + 7·y + 6·z
The number of minutes each exercise should be performed are;
Running = 44 minutes
Rowing = 15 minutes
Aerobics = 1 minute
The reason for arriving at the above values is as follows:
The given parameters are:
The total duration of the workout = 60 minutes
The calories burnt per minute by running = 9 calories
Calories burnt per minute by performing aerobics = 6 calories
Calories burnt per minute by rowing = 7 calories
The number of exercises to be performed = The three exercises
The duration of the time for running, x ≥ 5 minutes
Duration of the time for rowing, y ≥ 15 minutes
Duration of the time for aerobics, z ≤ 30 minutes
The system of inequalities based on the constraints are;
x + y + z = 60
x ≥ 5
y ≥ 15
0 < z ≤ 30
Objective function
The objective is to find the duration of each exercise that result in burning the maximum number of calories
Therefore, the objective function is the function that gives the amount of calories burnt, which is the sum of the product of the calorie burnt per minute for a given exercise and the duration of the exercise
The objective function is, f(x, y, z) = 9·x + 7·y + 6·z
Calculating the number of minutes for performing each exercise to burn the maximum calories:
Running burns the most calories, to burn maximum calories, we have;
Running duration = 60 mins - (Minimum duration aerobics + Minimum duration running)
Aerobics burns the least calories
∴ Minimum duration aerobics, z = 1 minute (minimum value possible)
Rowing duration is at least 15 minutes
∴ Minimum duration running, y = 15 minutes
∴ Running duration, x = 60 - (1 + 15) = 44
Running duration, x = 44 minutes
To perform all three exercises and burn maximum calories;
Running = 44 minutes
Rowing = 15 minutes
Aerobics = 1 minute
f(44, 16, 1) = 44 × 9 + 15 × 7 + 1 × 6 = 507
The maximum calories burnt, f(44, 16, 1) = 507 calories
Hence, using objective function and linear inequalities in linear programing problem, 44 minutes of running, 16 minutes of rowing and 1 minutes of aerobics should you perform each exercise to burn the maximum of 507 calories.
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