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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. And your aerobics session should be no more than 30 minutes. How many minutes should you perform each exercise to burn the maximum calories? Write a system of inequalities based on the given constraints. Then note the objective function

Respuesta :

apologiabiology
running burns the most calories folwed by aerobic

so max out the most and min the least


most running
running has no limit, so see least

aerobic, aerobic has to be at most 30 (less than 30)

rowing has to be at least 15 mins
say row=15 mins

then you have 45 mins left
max out on running (minus 1 tho to do running)

so
44mins running 1 min run, 15min row
total calories=507 calories burned

the way to do it is
x=running minutes
y=aerobic minutes
z=rowing minutes

x+y+z=60
9x+6y+7z=max calories
x≥5
y≤30
z≥15
those are the equations
not sure what the objective function is
  • The system of inequality is;

     [tex]{}[/tex]   x + y + z = 60

       [tex]{}[/tex] x ≥ 5

        [tex]{}[/tex]y ≥ 15

        [tex]{}[/tex]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;

        [tex]{}[/tex]Running = 44 minutes

   [tex]{}[/tex]     Rowing = 15 minutes

[tex]{}[/tex]         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

Learn more about objective function here;

https://brainly.com/question/13027008

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