Problem 55 - Precalculus Mathematics for Calculus

Question

Nutrition A doctor recommends that a patient take \(50 \mathrm{mg}\) each of niacin, riboflavin, and thiamin daily to alleviate a vitamin deficiency. In his medicine chest at home the patient finds three brands of vitamin pills. The amounts of the relevant vitamins per pill are given in the table. How many pills of each type should he take every day to get \(50 \mathrm{mg}\) of each vitamin? $$\begin{array}{|l|c|c|c|} \hline & \text { VitaMax } & \text { Vitron } & \text { VitaPlus } \\ \hline \text { Niacin (mg) } & 5 & 10 & 15 \\ \text { Riboflavin (mg) } & 15 & 20 & 0 \\ \text { Thiamin (mg) } & 10 & 10 & 10 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

2 VitaMax, 1 Vitron, 2 VitaPlus pills daily.

Step 1: Define Variables

Let \( x \), \( y \), and \( z \) represent the number of VitaMax, Vitron, and VitaPlus pills, respectively, that the patient should take.

Step 2: Set Up System of Equations

Use the nutritional data from the table to construct equations for each vitamin: 1. Niacin: \( 5x + 10y + 15z = 50 \)2. Riboflavin: \( 15x + 20y + 0z = 50 \)3. Thiamin: \( 10x + 10y + 10z = 50 \)

Step 3: Simplify the Thiamin Equation

Simplify the third equation: \( 10x + 10y + 10z = 50 \) becomes \( x + y + z = 5 \).

Step 4: Solve the System of Equations - Substitution Method

Start with the simplified thiamin equation: \( x + y + z = 5 \).Substitute \( z = 5 - x - y \) into the niacin and riboflavin equations.

Step 5: Solve Niacin Equation with Substitution

Substitute \( z = 5 - x - y \) into \( 5x + 10y + 15z = 50 \):\[ 5x + 10y + 15(5 - x - y) = 50 \]\[ 5x + 10y + 75 - 15x - 15y = 50 \]\[ -10x - 5y = -25 \]\[ 2x + y = 5 \]

Step 6: Solve Riboflavin Equation with Substitution

Substitute \( z = 5 - x - y \) into \( 15x + 20y = 50 \):This equation remains unchanged as there's no \( z \) term.So we have: \( 15x + 20y = 50 \)

Step 7: Solve the New System of Equations

Now solve:1. \( 2x + y = 5 \)2. \( 15x + 20y = 50 \)From the first equation, express \( y = 5 - 2x \) and substitute into the second:\[ 15x + 20(5 - 2x) = 50 \]\[ 15x + 100 - 40x = 50 \]\[ -25x = -50 \]\[ x = 2 \]Then substitute \( x = 2 \) back into \( y = 5 - 2x \):\[ y = 5 - 2(2) = 1 \]

Step 8: Determine \( z \) Value

Use the equation \( x + y + z = 5 \) with \( x = 2 \) and \( y = 1 \): \[ z = 5 - 2 - 1 = 2 \]

Step 9: Conclusion

The patient should take 2 VitaMax pills, 1 Vitron pill, and 2 VitaPlus pills daily to meet the required vitamin intake.

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with a common set of variables. In the context of vitamin deficiency, this helps solve problems where multiple factors, like different vitamin pill brands, contribute to a nutritional goal. By expressing the amounts of niacin, riboflavin, and thiamin in each vitamin pill as equations, we can use algebraic methods to find a combination of pills that meet the daily prescribed intake. In our example, the equations are:

• Niacin: \( 5x + 10y + 15z = 50 \)
• Riboflavin: \( 15x + 20y = 50 \)
• Thiamin: \( x + y + z = 5 \)

Establishing these equations is the first step in determining how many pills of each type—VitaMax, Vitron, and VitaPlus—the patient should take daily.

Nutritional Data

Nutritional data provides key information about the content of vitamins and minerals in various food or supplement items. In our exercise, the nutritional data of each vitamin pill brand is presented in a table, outlining the amounts of niacin, riboflavin, and thiamin they contain. This data is crucial for setting up the system of equations needed to solve for the number of each pill type required to meet the dietary recommendations. Knowing how each pill contributes to the total intake, we can use this information to build precise equations for problem-solving. This data guides the entire process, ensuring that the combination of pills gives exactly the right amount of vitamins needed to combat deficiency.

Algebraic Substitution

Algebraic substitution is a method used in solving systems of equations where one variable is expressed in terms of others and replaced in subsequent equations. This simplifies the system, making it easier to solve. In our example, the thiamin equation is simplified to \( x + y + z = 5 \) and rearranged to express \( z \) as:

• \( z = 5 - x - y \)

By substituting \( z = 5 - x - y \) into the niacin and riboflavin equations, the system reduces the number of variables, making it more manageable to find the exact values of \( x \), \( y \), and \( z \), which represent the number of pills required.

Vitamins and Minerals

Vitamins and minerals are essential nutrients the body needs to function correctly. They play a crucial role in maintaining health and preventing deficiencies. In this exercise, the focus is on three specific vitamins: niacin, riboflavin, and thiamin. Each has unique roles:

• Niacin helps in converting food into energy and is important for the nervous system.
• Riboflavin is required for energy production and cellular function.
• Thiamin plays a key role in energy metabolism and is essential for the proper functioning of muscles, nerves, and the heart.

Proper intake of these vitamins ensures the body's biochemical processes run smoothly, highlighting why the doctor prescribed a daily intake of 50 mg each to the patient.

Algebraic Problem Solving

Algebraic problem-solving involves using mathematical techniques to find unknown values within given parameters. In the vitamin deficiency problem, solving the system of equations is key to determining how many of each vitamin pill the patient should take. Initially, equations are set and simplified using substitution to handle them more easily. With each step, simplify and manipulate the equations until all unknowns are solved, ensuring each pill fulfills the required vitamin amounts. Thus, algebra provides a structured approach to deciphering complex scenarios involving multiple variables and constraints, leading to practical, real-world solutions.