Problem 27 - A Graphical Approach to College Algebra

Question

Exercises 27 and 28 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

Approximately 266.67 gallons of 94-octane gasoline should be used.

Step 1: Understand the problem

We need to find how many gallons of 94-octane gasoline must be added to 400 gallons of 99-octane gasoline to create a mixture that is 97-octane. This involves the concept of weighted averages.

Step 2: Set up the equation

Denote the number of gallons of 94-octane gasoline needed as \(x\). The mixture will thus have a total volume of \(x + 400\) gallons. The 94-octane and 99-octane gasolines contribute a certain percentage of isooctane to the mixture. The goal is for the total mixture to be 97% isooctane.

Step 3: Write the mixture equation

The contribution of isooctane from 94-octane gasoline is \(0.94x\) gallons and from 99-octane gasoline is \(0.99 \times 400\) gallons. We want the total isooctane amount to equal \(0.97(x + 400)\) gallons: \[ 0.94x + 0.99 \times 400 = 0.97(x + 400) \]

Step 4: Simplify and solve the equation

Expand and simplify the equation: \[ 0.94x + 396 = 0.97x + 388 \] Rearrange terms to group \(x\) on one side: \[ 396 - 388 = 0.97x - 0.94x \] This simplifies to: \[ 8 = 0.03x \] Finally, solve for \(x\) by dividing both sides by 0.03: \[ x = \frac{8}{0.03} = 266.67 \]

Step 5: Verify the solution

Plug \(x = 266.67\) back into the context to ensure it makes sense. The total gallons of gasoline become \(266.67 + 400 = 666.67\) gallons. The isooctane contribution is: 94% of 266.67 plus 99% of 400, which equals 97% of 666.67, confirming our setup and calculation is accurate.

Key Concepts

Octane Rating

Octane rating is a measure of a fuel's ability to resist knocking or pinging during combustion, caused by the air-fuel mixture within an engine cylinder detonating prematurely. The octane number signifies how well a particular fuel mix will perform under pressure. In simpler terms, it's a standard that indicates how smoothly fuel burns. When referring to a gasoline blend, the octane rating tells us the percentage of isooctane in a fuel mixture. For example, a 98-octane gasoline mix behaves like a standard fuel containing 98% isooctane and 2% heptane.
This becomes critical when combining different gasolines to achieve a desired performance level. Understanding octane ratings helps ensure a vehicle functions at optimal efficiency without engine damage.
In practical terms, selecting the correct octane level is about balancing between performance needs and cost. Higher octane fuels are typically used in high-performance engines. However, using them in standard engines often yields no added benefit.

Mixture Problems

Mixture problems involve combining substances to achieve a desired mix with specific properties. These problems often use weighted averages to find the correct proportions of each component. In the context of octane rating, we're blending fuels of different octane numbers. The goal is to create a new mixture with a specified octane rating.
In the problem, we have two fuels: 94-octane and 99-octane. We need to find how much 94-octane fuel will mix with a given amount of 99-octane to make a mixture that has a 97-octane rating. The weighted average is calculated by considering the contribution of each component relative to the total amount. Hence, the contributions of each type of gasoline should precisely blend, so that the mixture achieves the target octane rating.

• Establish variables for the unknown quantities.
• Determine how each component contributes to the total property of the mixture.
• Set up an equation representing these relationships.

Effective problem-solving in mixtures involves strategic planning and careful arithmetic to meet the desired conditions.

Algebraic Equations

Solving mixture problems often involves setting up algebraic equations. To solve for unknown quantities, like how many gallons of one type of gasoline are needed, we use equations to express relationships between the components roughly and mathematically.
Consider the equation from the exercise: \[ 0.94x + 0.99 \times 400 = 0.97(x + 400) \]Here, we see how the isooctane contribution from each fuel type relates to the mixture's desired property. By isolating the variable, we can solve for the unknown quantity, in this case, the number of gallons of 94-octane gasoline needed. Here's how:

• Simplify the equation by expanding it.
• Rearrange terms to isolate the variable \(x\).
• Solve the simplified equation for \(x\).

Through algebra, we solidify our understanding of how to adjust variable factors to achieve precise conditions. This example demonstrates the power of algebra in solving real-world problems, revealing the solution to be approximately 266.67 gallons of 94-octane gasoline needed to achieve the desired octane mixture.