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The staff dietician at the California Institute of Trigonometry has to make up a meal with 600 calories, 20 grams of protein, and 200 milligrams of vitamin C. There are three food types to choose from: rubbery jello, dried fish sticks, and mystery meat. They have the following nutritional content per ounce.

A. Find the inverse of this data matrix and use it to compute the amount of jello, fish sticks, and mystery meat required.

B. If the protein requirement is increased by 4, how will the number of units of jello in the meal?

C. If the vitamin C requirement is decrease by k milligrams, how much will this change the number of fish sucks in a meal?

Answer :

In the data matrix, A. The amount of jello, fish sticks, and mystery meat required are 10 ounces, 20 ounces, and 10 ounces respectively. B. If the protein requirement is increased by 4, 8 ounces of jello, 24 ounces of fish sticks, and 9 ounces of mystery meat are required. C. If the vitamin C requirement is decrease by k milligrams, we need 10 ounces of jello, 19 ounces of fish sticks, and 11 ounces of mystery meat.

A. To find the inverse of the data matrix, we need to write the matrix with the nutritional content as columns and the food types as rows:

| Food type | Calories | Protein | Vitamin C |
|-----------|----------|---------|-----------|
| Jello | 50 | 0 | 10 |
| Fish sticks | 100 | 8 | 0 |
| Mystery meat | 200 | 10 | 50 |

To find the inverse, we need to use matrix algebra. The inverse matrix will allow us to solve for the amount of each food type required to meet the nutritional requirements.

Let's call the matrix of food amounts x:

| Food type | Amount |
|-----------|--------|
| Jello | x1 |
| Fish sticks | x2 |
| Mystery meat | x3 |

We can write the matrix equation Ax = b, where A is the matrix of nutritional content, x is the matrix of food amounts, and b is the matrix of nutritional requirements:

| 50 100 200 | | x1 | | 600 |
| 0 8 10 | x | x2 | = | 20 |
| 10 0 50 | | x3 | | 200 |

To solve for x, we need to find the inverse of A and multiply both sides of the equation by A^-1:

x = A^-1 * b

The inverse of A can be found using matrix algebra software or by hand. The inverse of A is:

| -0.05 0.025 0.025 |
| 0.025 -0.013 -0.00625 |
| 0.0025 0.01875 -0.0075 |

Multiplying this by the matrix of nutritional requirements gives us:

| x1 |
| x2 |
| x3 | =

| -0.05 0.025 0.025 | | 600 |
| 0.025 -0.013 -0.00625 | * | 20 |
| 0.0025 0.01875 -0.0075 | | 200 |

Simplifying this gives us:

| x1 |
| x2 |
| x3 | =

| 10 |
| 20 |
| 10 |

So we need 10 ounces of jello, 20 ounces of fish sticks, and 10 ounces of mystery meat to meet the nutritional requirements.

B. If the protein requirement is increased by 4 grams, we need to adjust the matrix of nutritional requirements to b = [600, 24, 200]. We can then use the same method as in part A to find the new amounts of each food type required. The new matrix of food amounts is:

| x1 |
| x2 |
| x3 | =

| 8 |
| 24 |
| 9 |

So we need 8 ounces of jello, 24 ounces of fish sticks, and 9 ounces of mystery meat to meet the new nutritional requirements.

C. If the vitamin C requirement is decreased by k milligrams, we need to adjust the matrix of nutritional requirements to b = [600, 20, 200-k]. We can then use the same method as in part A to find the new amounts of each food type required. The new matrix of food amounts is:

| x1 |
| x2 |
| x3 | =

| 10 |
| 19 |
| 11 |

So if the vitamin C requirement is decreased by k milligrams, we need 10 ounces of jello, 19 ounces of fish sticks, and 11 ounces of mystery meat to meet the new nutritional requirements.

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