Problem 28 - A Graphical Approach to Precalculus with Limits

Question

involve octane rating of gasoline, \(a\) measure of its antiknock qualities. In one measure of octane, a standard fuel is made with only two ingredients: heptane and isooctane. For this type of fuel, the octane rating is the percent of isooctane. An actual gasoline blend is then compared with a standard fuel. For example, a gasoline with an octane rating of 98 has the same antiknock properties as a standard fuel that is \(98 \%\) isooctane. Octane Rating of Gasoline How many gallons of 94-octane gasoline should be mixed with 400 gallons of 99-octane gasoline to obtain a mixture that is 97 -octane?

Step-by-Step Solution

Answer

Mix 267 gallons of 94-octane gasoline with 400 gallons of 99-octane gasoline.

Step 1: Define Variables

Let \( x \) represent the number of gallons of 94-octane gasoline, which we want to find. We know we are mixing this with 400 gallons of 99-octane gasoline, and the resulting mixture should be 97-octane.

Step 2: Write an Equation for Isooctane Content

The octane rating corresponds to the percentage of isooctane. Hence, the amount of isooctane in the 94-octane gasoline is \( 0.94x \), and in the 99-octane gasoline is \( 0.99 \times 400 = 396 \) gallons. The total isooctane should equal the percentage in the resulting mixture: \( 0.97(x + 400) \).

Step 3: Set Up the Equation

We create the equation for the total isooctane before and after mixing:\[ 0.94x + 396 = 0.97(x + 400) \]

Step 4: Simplify the Equation

Simplify and solve for \( x \):1. Distribute on the right-hand side: \( 0.97x + 388 \).2. Equation becomes: \( 0.94x + 396 = 0.97x + 388 \).3. Rearrange to isolate terms involving \( x \): \( 396 - 388 = 0.97x - 0.94x \).4. Simplify to: \( 8 = 0.03x \).

Step 5: Solve for x

To find \( x \), divide both sides by \( 0.03 \):\[ x = \frac{8}{0.03} = 266.67 \]

Step 6: Conclusion

Thus, approximately 267 gallons of 94-octane gasoline should be mixed with 400 gallons of 99-octane gasoline to achieve the desired 97-octane mixture.

Key Concepts

Isooctane Content

To understand how octane ratings are determined, we need to dive into the concept of isooctane content. Octane rating measures a fuel's resistance to knocking or pinging during combustion, which is a critical aspect of engine performance. It's a standard for evaluating gasoline. The rating is actually based on the percentage of isooctane present in the fuel.

Most commonly, gasoline mixtures are compared to a reference fuel composed entirely of heptane and isooctane. Isooctane, a relatively stable hydrocarbon, grants the fuel its high octane rating. When you see a gasoline labeled as, let's say, 98-octane, it means that it behaves with the antiknock qualities equivalent to a fuel mixture that contains 98% isooctane.

Understanding isooctane content is crucial because adjustments in it can directly affect the performance and efficiency of engines. A higher percentage of isooctane in the mixture usually means improved engine performance due to fewer knocks.

Algebraic Equation

An algebraic equation is a powerful tool for solving problems, like determining the composition of a gasoline blend.

In the case of mixing gasoline to achieve a certain octane rating, we rely on an equation to represent the mixture's isooctane content. This equation essentially balances the amount of isooctane in different components to match the desired octane rating of the final blend.

In our example, we define variables: let \( x \) be the unknown gallons of 94-octane gasoline we need, and 400 gallons of 99-octane gasoline are given. Their isooctane contents are represented as \( 0.94x \) and \( 0.99 \times 400 \) respectively. Then, using the desired octane rating of 97, we write the relationship as:
\[ 0.94x + 396 = 0.97(x + 400) \]
This equation states that the sum of isooctane from both gasoline types equals the expected isooctane percentage of the mixture. Solving this equation involves rearranging terms and finding values that balance both sides, arriving at the solution for \( x \), the unknown quantity of 94-octane gasoline needed.

Mixture Problems

Mixture problems are a common type of algebra problem that involves finding an unknown component in a mixture. This could be related to calculating concentrations, percentages, or proportions of substances.

In our exercise, a mixture problem is solved to determine how much of one type of gasoline is necessary to achieve a certain octane rating when mixed with another gasoline. This involves

• identifying the known quantities (in our case, the 400 gallons of 99-octane gasoline),
• assigning variables to the unknowns (like the \( x \) gallons of 94-octane gasoline), and
• writing an equation that represents the relationship.

Using algebra, we calculate the needed amount of one component to achieve the desired mixture characteristics. These problems often require careful conversion of percentages into decimal fractions and precise algebraic manipulation.

Mixture problems help in a variety of fields, including chemistry, cooking, engineering, and medicine, where mixing different ingredients results in a new product or solution. Understanding them is essential for anyone working with blends or mixtures in practical scenarios.