Problem 1 - Technical Mathematics with Calculus

Question

Treat the percents given in this exercise as exact numbers, and work to three significant digits. Two different mixtures of gasohol are available, one with \(5 \%\) alcohol and the other containing \(12 \%\) alcohol. How many gallons of the \(12 \%\) mixture must be added to 252 gal of the \(5 \%\) mixture to produce a mixture containing \(9 \%\) alcohol?

Step-by-Step Solution

Answer

336 gallons of the 12% mixture must be added.

Step 1: Understanding the Problem

We need to determine the amount of a 12% alcohol gasohol mixture to mix with 252 gallons of a 5% alcohol gasohol mixture to obtain a final gasohol mixture with 9% alcohol.

Step 2: Setting Up the Equation

Let the volume of the 12% mixture to be added be represented by x. The total volume of alcohol in the mixed solution is the sum of the alcohol in the 5% mixture and the alcohol in the 12% mixture. The equation setup is based on the percentages and volume of alcohol: current amount of alcohol + added amount of alcohol = total amount of alcohol in the final mixture.Mathematically represented as:0.05(252) + 0.12x = 0.09(252 + x)

Step 3: Solving for x

Now, solve the equation for x.0.05(252) + 0.12x = 0.09(252 + x)12.6 + 0.12x = 22.68 + 0.09xSubtract 0.09x from both sides:12.6 + 0.03x = 22.68Subtract 12.6 from both sides:0.03x = 10.08Divide by 0.03 to find x:x = 10.08 / 0.03x = 336

Key Concepts

Mixture Problems

When dealing with mixture problems, the goal is often to find the proportions of ingredients needed to achieve a specific concentration in a final mixture. This can be encountered across various real-world applications, including compounding medications, blending coffee, or, as in our example, creating gasohol—a blend of gasoline and alcohol.

The core principle is to maintain a balance between the quantities and concentrations of the mixtures being combined. To solve mixture problems, you set up an equation that equates the amount of the substance of interest (in this case, alcohol) before and after the mixing process. By solving these balance equations, you can find the unknown quantity, which represents the amount to be added or removed to reach the desired final mixture.

Percentage Concentration

Percentage concentration is a way of expressing the amount of a substance in a mixture as a percentage of the total mixture. It is commonly used in chemistry to indicate the strength or concentration of a solution. For example, a mixture with a `5%` concentration of alcohol contains 5% alcohol by volume and 95% other substances—typically gasoline in the case of gasohol.

Understanding how to work with percentage concentrations is vital for correctly setting up mixture problems. In our gasohol example, different percentages (5%, 12%, and 9%) represent the proportion of alcohol in the original mixtures and the desired final mixture. By converting these percentages into decimals, we can use them in algebraic equations to help us solve the problem at hand.

Algebraic Equations

Algebraic equations are mathematical statements that assert the equality of two expressions. They are essential tools used to solve mixture problems by setting up relationships between different variables. These equations can often contain one or more unknowns, which we are required to solve for.

In mixture problems, we usually use algebra to represent the relationships between the volumes and concentrations of the substances being mixed. For instance, as in the given problem about gasohol, we use the equation \(0.05(252) + 0.12x = 0.09(252 + x)\) to relate the volumes of the different gasohol mixtures according to their concentrations. By manipulating this equation and simplifying, we isolate the variable \(x\) to solve for it.

Significant Digits

Significant digits (also known as significant figures) are the digits in a number that carry meaning contributing to its precision. They include all the non-zero numbers, any zeros between them, and any trailing zeros in the decimal portion.

In scientific measurements and calculations, maintaining the correct number of significant digits is crucial for accuracy and indicates the quality of the measurement. For the gasohol problem, the requirement to work to three significant digits ensures that the answer reflects the precision of the original measurements. This practice prevents the overestimation of the precision of calculated results and maintains consistency in reporting numeric answers.