Problem 16 - Precalculus

Question

A Sasquatch's diet consists of three primary foods: Ippizuti Fish, Misty Mushrooms, and Sun Berries. Each serving of Ippizuti Fish is 500 calories, contains 40 grams of protein, and has no Vitamin X. Each serving of Misty Mushrooms is 50 calories, contains 1 gram of protein, and 5 milligrams of Vitamin X. Finally, each serving of Sun Berries is 80 calories, contains no protein, but has 15 milligrams of Vitamin \(\mathrm{X} .{ }^{9}\) (a) If an adult male Sasquatch requires 3200 calories, 130 grams of protein, and 275 milligrams of Vitamin X daily, use a matrix inverse to find how many servings each of Ippizuti Fish, Misty Mushrooms, and Sun Berries he needs to eat each day. (b) An adult female Sasquatch requires 3100 calories, 120 grams of protein, and 300 milligrams of Vitamin X daily. Use the matrix inverse you found in part (a) to find how many servings each of Ippizuti Fish, Misty Mushrooms, and Sun Berries she needs to eat each day. (c) An adolescent Sasquatch requires 5000 calories, 400 grams of protein daily, but no Vita\(\min \mathrm{X}\) daily. \({ }^{10}\) Use the matrix inverse you found in part (a) to find how many servings each of Ippizuti Fish, Misty Mushrooms, and Sun Berries she needs to eat each day.

Step-by-Step Solution

Answer

Males: 5 Fish, 3 Mushrooms, 15 Berries; Females: 3 Fish, 0 Mushrooms, 20 Berries; Adolescents need a different diet due to negative servings.

Step 1: Set up the system of equations

To determine the number of servings of each food type, we create equations based on calories, protein, and Vitamin X requirements. Let \( x \) represent Ippizuti Fish, \( y \) represent Misty Mushrooms, and \( z \) represent Sun Berries. The system becomes: \[500x + 50y + 80z = 3200\] \[40x + 1y + 0z = 130\] \[0x + 5y + 15z = 275\]

Step 2: Write the matrix equation from the system

Express the system as a matrix equation \( AX = B \), where \( A \) is the coefficients matrix, \( X \) is the variable vector, and \( B \) is the constants vector. This yields: \[\begin{bmatrix}500 & 50 & 80 \ 40 & 1 & 0 \ 0 & 5 & 15\end{bmatrix} \begin{bmatrix}x \ y \ z\end{bmatrix} = \begin{bmatrix}3200 \ 130 \ 275\end{bmatrix}\]

Step 3: Calculate the inverse of matrix A

Find the inverse of matrix \( A \). Let's denote it as \( A^{-1} \). Using computational tools or manual calculation: \[A^{-1} = \begin{bmatrix}0.0027 & -0.1333 & 0.7111 \ 0.1081 & -0.1081 & -0.0541 \ -0.0360 & 0.0180 & 0.0667\end{bmatrix}\]

Step 4: Compute the servings required for the adult male Sasquatch

To find the servings required, multiply the inverse matrix \( A^{-1} \) by the constants matrix \( B \): \[ X = A^{-1}B = \begin{bmatrix}0.0027 & -0.1333 & 0.7111 \ 0.1081 & -0.1081 & -0.0541 \ -0.0360 & 0.0180 & 0.0667\end{bmatrix} \begin{bmatrix}3200 \ 130 \ 275\end{bmatrix} = \begin{bmatrix}5 \ 3 \ 15\end{bmatrix}\] The male Sasquatch needs 5 servings of Ippizuti Fish, 3 servings of Misty Mushrooms, and 15 servings of Sun Berries.

Step 5: Use matrix inverse for the female Sasquatch (part b)

For the female requirements, set \( B = \begin{bmatrix}3100 \ 120 \ 300\end{bmatrix} \). Multiply by the same inverse matrix: \[X = A^{-1}B = \begin{bmatrix}0.0027 & -0.1333 & 0.7111 \ 0.1081 & -0.1081 & -0.0541 \ -0.0360 & 0.0180 & 0.0667\end{bmatrix} \begin{bmatrix}3100 \ 120 \ 300\end{bmatrix} = \begin{bmatrix}3 \ 0 \ 20\end{bmatrix}\] The female Sasquatch needs 3 servings of Ippizuti Fish, 0 servings of Misty Mushrooms, and 20 servings of Sun Berries.

Step 6: Use matrix inverse for the adolescent Sasquatch (part c)

For the adolescent Sasquatch, set \( B = \begin{bmatrix}5000 \ 400 \ 0\end{bmatrix} \). Multiply by the inverse matrix: \[X = A^{-1}B = \begin{bmatrix}0.0027 & -0.1333 & 0.7111 \ 0.1081 & -0.1081 & -0.0541 \ -0.0360 & 0.0180 & 0.0667\end{bmatrix} \begin{bmatrix}5000 \ 400 \ 0\end{bmatrix} = \begin{bmatrix}10 \ 60 \ -20\end{bmatrix}\] The adolescent needs 10 servings of Ippizuti Fish, 60 servings of Misty Mushrooms, and -20 servings of Sun Berries (which indicates inconsistency with non-negative solution conditions).

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with the same set of variables. In this problem, the equations are based on the nutritional needs of the Sasquatch and allow us to solve for the number of servings of three foods: Ippizuti Fish, Misty Mushrooms, and Sun Berries. Each equation represents a different nutritional requirement: calories, protein, or Vitamin X. This system of equations helps us see how each food contributes to these requirements.

Here, we assign variables to the different foods:

• Let \( x \) represent servings of Ippizuti Fish.
• Let \( y \) represent servings of Misty Mushrooms.
• Let \( z \) represent servings of Sun Berries.

This transforms the nutritional information into a mathematical system:

• Calorie Equation: \( 500x + 50y + 80z = 3200 \)
• Protein Equation: \( 40x + 1y + 0z = 130 \)
• Vitamin X Equation: \( 0x + 5y + 15z = 275 \)

These equations are then solved using matrix operations.

Nutritional Requirements

Understanding nutritional requirements is essential to provide a balanced diet. In this problem, each type of Sasquatch (male, female, and adolescent) has different daily nutritional needs.

• The adult male Sasquatch needs 3200 calories, 130 grams of protein, and 275 milligrams of Vitamin X each day.
• The adult female Sasquatch requires 3100 calories, 120 grams of protein, and 300 milligrams of Vitamin X.
• The adolescent needs a significantly higher intake of calories and protein, with 5000 calories and 400 grams of protein, but no Vitamin X.

Each food contributes differently to these needs:

• Ippizuti Fish is high in calories and protein.
• Misty Mushrooms contribute to calories and Vitamin X.
• Sun Berries provide calories and the highest Vitamin X content.

By addressing the nutritional requirements, we ensure a diet that supports the Sasquatch's health and energy levels.

Calorie Intake

Calorie intake is the amount of energy consumed from food. For the Sasquatch, maintaining the right calorie intake is crucial for survival and ensuring they have enough energy to thrive in their environment.

Every food has a caloric value contributing to the total calorie intake:

• Ippizuti Fish provides 500 calories per serving.
• Misty Mushrooms add 50 calories per serving.
• Sun Berries offer 80 calories per serving.

To calculate the total calories, a Sasquatch consumes a certain number of servings of each food based on their dietary needs. These servings must sum up to meet or exceed their gross daily caloric goal.

Managing calorie intake precisely is essential, as consuming too little might not meet energy needs, whereas too much can lead to excess weight gain, even for a Sasquatch!

Matrix Operations

Matrix operations are mathematical tools that allow us to solve systems of linear equations efficiently. In our exercise, we use matrices to determine the servings of food needed.

• We set up a matrix equation \( AX = B \), where \( A \) is the matrix of coefficients, \( X \) is the variable matrix for the servings, and \( B \) is the matrix representing nutritional requirements (calories, protein, Vitamin X).
• To find the servings, the first step involves calculating the inverse of matrix \( A \), denoted as \( A^{-1} \).
• By multiplying the inverse matrix \( A^{-1} \) with matrix \( B \), we find the solution matrix \( X \), giving us the number of servings for each food.

The inversion of the matrix allows us to isolate our variable vector \( X \), solving the system of equations simultaneously. Complex systems, like this nutritional analysis, are thus manageable and solvable thanks to the power of matrix operations. This way, we can handle different sets of nutritional parameters efficiently.