Problem 44 - Precalculus with Limits

Question

Nutrition One eight-ounce glass of apple juice and one eight-ounce glass of orange juice contain a total of 179.2 milligrams of vitamin \(\mathrm{C}\) . Two eight-ounce glasses of apple juice and three eight-ounce glasses of orange juice contain a total of 442.1 milligrams of vitanin C. How much vitamin \(C\) is in an eight-ounce glass of each type of juice?

Step-by-Step Solution

Answer

An eight-ounce glass of apple juice contains 103.5 milligrams of vitamin C, and an eight-ounce glass of orange juice contains 75.7 milligrams of vitamin C.

Step 1: Translate problem to mathematical form

The first equation according to the problem is \(A + O = 179.2\) where A and O represent the Vitamin C content in Apple juice and Orange juice respectively. The second equation is \(2A + 3O = 442.1\). Now that we've translated the problem into mathematical form, the next step would be to solve these equations.

Step 2: Solve equations using substitution or elimination

Since the first equation involves only one glass of each kind, we can use substitution. Solve the first equation for A: \(A = 179.2 - O\). Substitute \(A\) in the second equation yields a new equation: \(2(179.2 - O) + 3O = 442.1\). Solving for \(O\), we get \(O = 75.7\) milligrams.

Step 3: Substitute \(O = 75.7\) into equation one

Substitute the value of \(O\) in the first equation: \(A + 75.7 = 179.2\). Solving for \(A\) gives us \(A = 103.5\) milligrams.

Key Concepts

Linear Equations

When we talk about linear equations, we refer to mathematical statements where two expressions are set equal to each other, and they graphically represent a straight line. In the context of algebra word problems, linear equations are used to describe relationships between different quantities.

For instance, let's consider the vitamin C problem where the total milligrams of vitamin C in glasses of juice is given. Here, two linear equations are created to depict the total vitamin C in terms of the number of glasses. The standard form of a linear equation is expressed as ax + by = c, where a, b, and c are constants, and x and y are variables representing the quantities we're interested in.

In our juice problem, A and O are the variables representing the amount of vitamin C in apple and orange juice respectively. This illustrates the power of linear equations in modeling and solving real-world scenarios.

Substitution Method

The substitution method is a technique used to solve systems of equations where one equation in the system is solved for one variable in terms of the other variables. This value is then substituted into the other equations.

This method is particularly useful when equations are already solved for a variable, or can be easily rearranged to solve for a variable. In the vitamin C example, one equation is rearranged to solve for A, which is then substituted into another equation to find the value of O. After which, the known value of O is substituted back into the original equation to find the value of A.

It's a sequential process that simplifies the system to one variable at a time, thus making it easier to solve complex problems with multiple unknowns.

Algebra Word Problems

Algebra word problems require translation of a verbal description into algebraic language—this means into equations that represent the relationships described in the problem.

One critical step is defining variables to represent the quantities in question, such as assigning A for the amount of vitamin C in an apple juice glass and O for the orange juice glass. It then involves creating equations that reflect the scenarios presented, like the total amount of vitamin C for different combinations of juice glasses.

Solving these word problems often involves systematic steps: interpreting the problem, translating into equations, using algebraic techniques (like the substitution method), and then interpreting the algebraic solution in the context of the original problem. With practice, these steps can make even complex real-world issues solvable through algebra.