Problem 108 - Precalculus Essentials

Question

You will be developing functions that model given conditions. A chemist working on a flu vaccine needs to mix a \(10 \%\) sodium-iodine solution with a \(60 \%\) sodium-iodine solution to obtain a 50 -milliliter mixture. Write the amount of sodium iodine in the mixture, \(S,\) in milliliters, as a function of the number of milliliters of the \(10 \%\) solution used, \(x\). Then find and interpret \(S(30)\)

Step-by-Step Solution

Answer

The amount of sodium iodine in the mixture when 30 milliliters of the 10% solution is used, is 15 milliliters.

Step 1: Understand the problem

First, recognize that you are mixing together two solutions with different percentages of sodium iodine, and that together, they add up to a total of 50 milliliters. The 10% solution contributes \(0.1x\) milliliters of sodium iodine (where \(x\) is the number of milliliters of the 10% solution used), and the 60% solution contributes \(0.6(50-x)\) milliliters of sodium iodine (as the remaining \(50-x\) milliliters of the mixture is comprised of this solution).

Step 2: Formulate the equation

The total amount of sodium iodine in the 50-milliliter mixture is the sum of the amounts contributed by each of the two solutions. Therefore, the function we are looking for is \(S(x) = 0.1x + 0.6(50 - x)\).

Step 3: Simplify the equation

We can simplify the equation for \(S(x)\) by carrying out the multiplication in the second term: \(S(x) = 0.1x + 30 - 0.6x\). Combine the x terms to get the final form of the function: \(S(x) = -0.5x + 30\).

Step 4: Find S(30)

Substitute \(x = 30\) into the function to find \(S(30)\). This gives \(S(30) = -0.5*30 + 30 = 15\) milliliters.

Step 5: Interpret the result

The result S(30) = 15 means that if we use 30 milliliters of the 10% solution, the final 50-milliliter mixture will contain 15 milliliters of sodium iodine.

Key Concepts

Percent Concentration

Understanding percent concentration is crucial when dealing with mixture problems. In the context of our exercise, concentration refers to the amount of a specific substance (sodium iodine) present in a solution relative to the total volume of the mixture. When we say a solution has a '10% sodium-iodine concentration,' it means that 10% of this solution's volume is pure sodium iodine. Thus, for every 100 milliliters of this solution, there are 10 milliliters of sodium iodine.

To visualize the concentration calculations, consider a situation where you have a total of 50 milliliters of a new mixture. If you used 30 milliliters of the 10% solution, you're effectively adding 3 milliliters of sodium iodine to the mix, since 10% of 30 is 3. It's important to grasp this principle, as it lays the groundwork for understanding the function modeling aspect of the solution.

Function Modeling

Function modeling is a mathematical tool for representing real-world relationships through equations. In our textbook problem, the amount of sodium iodine in the mixture, denoted as the function S, depends on the volume of the 10% solution used, represented by the variable x. This type of function is commonly known as a dependent variable because its value is determined by the independent variable.

Developing the Function

The relationship between the variables in our mixture problem was formulated as the function: \( S(x) = 0.1x + 0.6(50 - x) \). This equation models the concentration dynamics of the mix, where the sodium iodine contribution from each of the two solutions is accounted for. It's useful to remember that function modeling can help to predict outcomes and analyze scenarios within defined parameters, which is a skill applicable in various disciplines beyond chemistry.

System of Equations

In more complex scenarios, if we're dealing with multiple unknowns or variables, we'd need a system of equations to find a solution. Although our current problem involves only a single equation, the concept is similar: we're solving for one variable (the number of milliliters of sodium iodine) in terms of another (the volume of the 10% solution used).

When using a system of equations, you would combine different linear equations to solve for multiple variables, often visualized graphically where the solution is the point or points where the equations intersect. Since we're dealing with a straightforward linear relationship in this exercise, we only needed to manipulate one equation to find our function S(x). Nonetheless, understanding how to work with multiple interdependent relationships is a fundamental skill in precalculus.

Real-World Applications

Our mixture problem is more than just an academic exercise; it illustrates how math can solve real-world problems. Chemists, like in our exercise, often have to calculate precise concentrations of solutions in their work. The ability to create mixtures of specific concentrations is critical in fields such as medicine, pharmaceuticals, and environmental science.

In these fields, function modeling and system of equations play a significant role. They help to predict how changing one component of a system (like the volume of a solution) affects another (like concentration of the active ingredient). Whether it's determining the required dosage of a medication or calculating the dilution of pollutants, the skills exemplified in this mixture problem are directly transferable to practical solutions that impact our health and environment.