Problem 1 - Algebra and Trigonometry

Question

Use the model given to answer the questions about the object or process being modeled. The gas mileage \(M\) (in mi/gal) of a car is modeled by \(M=N / G\) where \(N\) is the number of miles driven and \(G\) is the number of gallons of gas used. (a) Find the gas mileage \(M\) for a car that drove 230 miles on 5.4 gallons of gas. (b) A car with a gas mileage \(M=25 \mathrm{mi} / \mathrm{gal}\) is driven 185 miles. How many gallons of gas are used?

Step-by-Step Solution

Answer

(a) 42.59 mi/gal, (b) 7.4 gallons.

Step 1: Identify Variables for Part (a)

For part (a), identify that \(N = 230\) miles and \(G = 5.4\) gallons. We need to find \(M\), the gas mileage.

Step 2: Apply the Gas Mileage Formula for Part (a)

Use the formula \(M = \frac{N}{G}\). Substitute \(N = 230\) and \(G = 5.4\) into the equation: \(M = \frac{230}{5.4}\).

Step 3: Calculate Gas Mileage for Part (a)

Perform the division: \(M = \frac{230}{5.4} \approx 42.59\) mi/gal.

Step 4: Identify Variables for Part (b)

For part (b), identify that \(M = 25\) mi/gal and \(N = 185\) miles. We need to find \(G\), the number of gallons used.

Step 5: Rearrange the Formula for Part (b)

Starting from \(M = \frac{N}{G}\), rearrange the equation to solve for \(G\): \(G = \frac{N}{M}\).

Step 6: Apply and Calculate for Part (b)

Substitute \(N = 185\) miles and \(M = 25\) mi/gal into the equation \(G = \frac{185}{25}\). Calculate \(G = \frac{185}{25} = 7.4\) gallons.

Key Concepts

Algebraic Modeling

When we talk about algebraic modeling, we are essentially discussing a mathematical approach to represent real-world situations using algebraic expressions or equations. In the context of calculating gas mileage, this involves expressing the relationship between variables like distance (miles driven), fuel consumption (gallons used), and mileage efficiency (miles per gallon) through an algebraic equation.
In our exercise, the equation provided is the model:

• Gas mileage, \( M \), is calculated using the formula \( M = \frac{N}{G} \).
• Here, \( N \) stands for the number of miles driven and \( G \) is the number of gallons of gas used.

This algebraic model allows us to plug in values for the variables we know to find the unknown variable. Using these equations can help students structure problem-solving processes and make complex problems more manageable by breaking them down into simple mathematical terms.

Division in Mathematics

Division is a fundamental operation in mathematics where you split a number into equal parts. It's a tool that allows us to find how many times one number is contained within another.
For gas mileage, division plays a key role. It helps find out how efficiently a car uses fuel:

• In part (a), to find the gas mileage \( M \), we divide the number of miles driven \( N \) (230 miles) by the gallons of fuel used \( G \) (5.4 gallons), which gives us \( M = \frac{230}{5.4} \).
• In part (b), we rearrange the equation to find the gallons used when the mileage \( M \) and miles driven \( N \) are known. Here, you divide 185 miles by 25 mi/gal, resulting in \( G = \frac{185}{25} \).

Through division, we can understand relationships between quantities, like how much mileage we get per gallon of fuel, thus demonstrating the practical applications of this operation in everyday life.

Problem Solving Steps

Effective problem-solving in mathematics involves a series of systematic steps. These steps ensure clarity in understanding the problem and accuracy in the solution.
Let’s break down the process using the gas mileage problem:

• **Identify Variables**: Recognize what you know and what you need to find. In part (a), you know the miles (230) and gallons (5.4), and you need \( M \). In part (b), you know \( M = 25 \) and miles (185), and need \( G \).
• **Use and Rearrange Formulas**: Apply the given model \( M = \frac{N}{G} \). If needed, rearrange it to find the unknown. For part (b), rearrange to \( G = \frac{N}{M} \).
• **Calculate**: Perform the necessary calculations. For example, calculate \( M \) and \( G \) based on the rearranged formulas given in parts (a) and (b).

By following these structured steps, students can methodically approach mathematical problems. This process promotes logical thinking and enhances problem-solving skills.