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A body builder is watching his intake of fat, carbs, and protein while training. He would like a meal consisting of a baked potato with butter, cottage cheese, and a chicken breast that contains 44.4 grams of fat, 48.6 grams of carbohydrates, and 131.5 grams of protein. An average baked potato with butter contains 11 grams of fat, 24 grams of carbs, and 3 grams of protein. A serving of cottage cheese contains 4.4 grams of fat, 8.2 grams of carbs, and 31.1 grams of protein. A serving of chicken breast contains 10.1 grams of fat, 0 grams of carbs, and 17.6 grams of protein. Which matrix represents the system of equations that will give the number of servings of each type of food the body builder should eat?

Respuesta :

Given:

  • The full meal should have 44.4g of fat, 48.6g of carbohydrates, 131.5g of protein
  • One serving of Baked Potato with Butter has 11g, 24g and 3g of fat, carbs and protein respectively.
  • One serving of cottage cheese has 4.4g, 8.2g and 31.1g of fat, carbs and protein respectively.
  • One serving of Chicken breast has 10.1g, 0g and 17.6g of fat, carbs and protein respectively.

To Find: The matrix representation of the system of equations that will give the number of servings of each type of food the bodybuilder should eat.

Solution:

[tex]\left[\begin{array}{cccc}11&4.4&10.1&44.4\\24&8.2&0&48.6\\3&31.1&17.6&131.5\end{array}\right][/tex]

Explanation:

We begin by forming the system of equations that will give the number of servings of each type of food the bodybuilder should eat so that he can reach his targeted intake of fats, carbs and protein.

Let x denote the number of servings of baked potato he should have, y denote the number of servings of cottage cheese he should have and z denote the number of servings of chicken breast he should have to reach his desired amount of fats, carbs and protein.

The first equation will then denote the number of servings of each food he should have to reach a total of 44.4g of fat.

So, we multiply the amount of fat in one serving of each dish with the number of servings of each dish to have. After summing these amounts, he will get the desired amount of 44.4g of fat.

In mathematical terms, [tex]11x+4.4y+10.1z=44.4[/tex]

Similarly, the second equation will denote the number of servings of each food he should have to reach a total of 48.6g of carbohydrate.

We obtain the equation [tex]24x+8.2y+0z=48.6[/tex]

Finally, we obtain the third equation to find the number of servings of each food to reach the total amount of protein.

We get [tex]3x+31.1y+17.6z=131.5[/tex]

To construct the matrix, we will write the equations below each other in a list as follows

[tex]11x+4.4y+10.1z=44.4[/tex]

[tex]24x+8.2y+0z=48.6[/tex]

[tex]3x+31.1y+17.6z=131.5[/tex]

Each of these equations has 4 terms (3 terms on the left hand side and 1 term on the right hand side). So, the matrix will be a 3x4 matrix.

After this, we write the coefficients of each term in the corresponding position of the matrix.

So our final matrix will be

[tex]\left[\begin{array}{cccc}11&4.4&10.1&44.4\\24&8.2&0&48.6\\3&31.1&17.6&131.5\end{array}\right][/tex]

Answer:

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Step-by-step explanation:

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