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Chapter 3: Problem 52

Nutrition According to the nutritional information on a package of Honey Nut Cheerios \(^{\bar{k}}\) brand cereal, each 1 -ounce serving of Cheerios contains 3 grams protein and 24 grams carbohydrates. \({ }^{16}\) Each half-cup serving of enriched skim milk contains 4 grams protein and 6 grams carbohydrates. a. I am planning a meal of cereal and milk and I want it to supply 26 grams of protein, and 78 grams of carbohydrates. How should I prepare my meal? b. If I require \(A\) grams of protein and \(B\) grams of carbohydrates, give a formula that tells me how many servings of milk and Cheerios to use.

Short Answer

Expert verified
To supply 26 grams of protein and 78 grams of carbohydrates, you should prepare your meal with \(2\) servings of Honey Nut Cheerios and \(5\) servings of enriched skim milk. To generalize the formula for A grams of protein and B grams of carbohydrates, use the following equations: Protein: \(3X + 4Y = A\) Carbohydrates: \(24X + 6Y = B\) Solve these equations for X and Y, where X is the number of servings of Cheerios and Y is the number of servings of milk needed to supply A grams of protein and B grams of carbohydrates.

Step by step solution

01

Define Variables

Let x be the number of 1-ounce servings of Cheerios and y be the number of half-cup servings of enriched skim milk. #Step 2 - Form Equations#
02

Create Equations for Protein and Carbohydrates

From given information, we can create two equations relating x and y to the desired amounts of protein and carbohydrates: Protein: \(3x + 4y = 26\) Carbohydrates: \(24x + 6y = 78\) #Step 3 - Solve the System of Equations#
03

Use Substitution or Elimination Method

We can solve these equations using either substitution or elimination method. In this case, we will use the elimination method: First, let's multiply both sides of the Protein equation by 2 to eliminate y when we sum both equations: New Protein equation: \(6x + 8y = 52\) Now, sum the two equations: New Protein equation + Carbohydrates equation: \((6x + 8y) + (24x + 6y) = 52 + 78\) Simplify: \(30x + 14y = 130\) As we wanted to eliminate y, we will subtract both the equations: New Protein equation - Carbohydrates equation: \((6x + 8y) - (24x + 6y) = 52 - 78\) Simplify: \(-18x + 2y = -26\) Now, we can divide by -2 to get a simple equation for y: \(9x - y = 13\) #Step 4 - Calculate Numbers of Servings#
04

Find Values of x and y

Now, we can use any of the equations to calculate the value of x: From the simplified Carbohydrates equation: \(24x + 6y = 78\) Substituting y from the simplified Protein equation: \(24x + 6(9x - 13) = 78\) By solving this equation, we get: \(24x + 54x - 78 = 78\) \(78x = 156\) \(x = 2\) Now, we can substitute the value of x back into the simplified Protein equation to find the value of y: \(9(2) - y = 13\) \(18 - y = 13\) \(y = 5\) Thus, 2 servings of Cheerios and 5 servings of enriched skim milk will supply 26 grams of protein and 78 grams of carbohydrates. #Step 5 - Generalize Formula#
05

Create Formula with A grams of Protein and B grams of Carbohydrates

Let X be the number of servings of Cheerios and Y be the number of servings of milk needed to supply A grams of protein and B grams of carbohydrates. From given information, we can create two equations relating X and Y to the desired amounts of protein and carbohydrates: Protein: \(3X + 4Y = A\) Carbohydrates: \(24X + 6Y = B\) Solving these equations for X and Y will give the number of servings needed to supply A grams of protein and B grams of carbohydrates. Note: This formula will only work if A and B can be achieved by the given nutritional facts of Cheerios and enriched skim milk.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Algebra
Linear algebra is a branch of mathematics that deals with vector spaces and linear mappings between them. In practical situations, such as nutrition problems, it manifests through systems of linear equations representing different constraints.

For instance, each food item's protein and carbohydrate content contributes to the overall nutritional value of a meal. By setting variables to represent the servings of each food item and constructing equations based on their nutritional content, we can use linear algebra to determine the precise combination of servings to meet dietary requirements.

The simplicity of linear algebra lies in its use of vectors and matrices which can represent and solve problems with several variables efficiently, such as the number of servings needed to achieve a certain nutritional profile.
Mathematical Modeling
Mathematical modeling is the process of translating a real-world scenario into mathematical terms, using variables, equations, functions, and other mathematical structures.

In dietary meal planning, the quantities of protein and carbohydrates needed can be 'modeled' by creating a system of linear equations, with each variable representing a particular food item's serving. This technique makes it possible to calculate optimal combinations of food items to meet nutritional goals.

For example, modeling the nutrients in Cheerios and skim milk involves setting equations that account for protein and carbohydrates in each serving. This model can then be used to determine how many servings of each are needed to meet a specified nutritional intake.
Elimination Method
The elimination method is a technique in linear algebra for solving systems of equations. This method involves adding or subtracting equations to eliminate one of the variables, facilitating the solving process.

In our nutrition problem, eliminating one variable from the set of equations simplifies the system to a single equation with one variable. By strategically multiplying the equations, we can align coefficients and subtract one equation from the other to cancel out a specific variable.

The elimination method is especially useful when dealing with larger systems, as it can significantly reduce the complexity and straightforwardly lead to a solution, as witnessed in determining the servings of Cheerios and milk needed for our dietary requirements.
Dietary Meal Planning
Dietary meal planning involves calculating and combining different food items to meet specific nutrition goals. It is often based on macronutrient content鈥攑roteins, carbohydrates, and fats鈥攔equired by an individual.

By applying linear algebra and mathematical modeling to meal planning, we can arrive at precise serving sizes to fulfill protein and carbohydrate requirements as outlined in our problem scenario. Moreover, by generalizing the solution, we create a formula that can be applied to a variety of dietary needs, providing flexibility and utility in meal planning.

In educational contexts, such problems also demonstrate how mathematical tools can be applied to everyday life, making abstract concepts concrete and practical.

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Most popular questions from this chapter

Microbucks Computer Company, besides having the stock mentioned in Exercise 49 , gets shipments of parts every month in the amounts of 100 processor chips, 1,000 memory chips, and 3,000 vacuum tubes at the Pom II factory, and 50 processor chips, 1,000 memory chips, and 2,000 vacuum tubes at the Pom Classic factory. a. What will the company's inventory of parts be after six months? b. When (if ever) will the company run out of one of the parts?

Find: (a) the optimal mixed row strategy; (b) the optimal mixed column strategy, and (c) the expected value of the game in the event that each player uses his or her optimal mixed strategy. $$ P=\left[\begin{array}{rr} -1 & 0 \\ 1 & -1 \end{array}\right] $$

Revenue Karen Sandberg, your competitor in Suburban State U's T-shirt market, has apparently been undercutting your prices and outperforming you in sales. Last week she sold 100 tie dye shirts for \(\$ 10\) each, 50 (low quality) Crew shirts at \(\$ 5\) apiece, and 70 Lacrosse T-shirts for \(\$ 8\) each. Use matrix operations to calculate her total revenue for the week.

If \(A\) and \(B\) are invertible, check that \(B^{-1} A^{-1}\) is the inverse of \(A B\).

Textbook Writing You are writing a college-level textbook on finite mathematics, and are trying to come up with the best combination of word problems. Over the years, you have accumulated a collection of amusing problems, serious applications, long complicated problems, and "generic" problems. \({ }^{25}\) Before your book is published, it must be scrutinized by several reviewers who, it seems, are never satisfied with the mix you use. You estimate that there are three kinds of reviewers: the "no-nonsense" types who prefer applications and generic problems, the "dead serious" types, who feel that a collegelevel text should be contain little or no humor and lots of long complicated problems, and the "laid-back" types, who believe that learning best takes place in a light-hearted atmosphere bordering on anarchy. You have drawn up the following chart, where the payoffs represent the reactions of reviewers on a scale of \(-10\) (ballistic) to \(+10\) (ecstatic): Reviewers ou \begin{tabular}{|l|c|c|c|} \hline & No-Nonsense & Dead Serious & Laid-Back \\ \hline Amusing & \(-5\) & \(-10\) & 10 \\ \hline Serious & 5 & 3 & 0 \\ \hline Long & \(-5\) & 5 & 3 \\ \hline Generic & 5 & 3 & \(-10\) \\ \hline \end{tabular} a. Your first draft of the book contained no generic problems, and equal numbers of the other categories. If half the reviewers of your book were "dead serious" and the rest were equally divided between the "no-nonsense" and "laid-back" types, what score would you expect? b. In your second draft of the book, you tried to balance the content by including some generic problems and eliminating several amusing ones, and wound up with a mix of which one eighth were amusing, one quarter were serious, three eighths were long, and a quarter were generic. What kind of reviewer would be least impressed by this mix? c. What kind of reviewer would be most impressed by the mix in your second draft?
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