Question

You are buying gasoline and notice that 14 gallons of gasoline is the same as 53 liters. (a) Use this information to find a mathematical model that relates gallons to liters. (b) Use the model to complete the table. $$\begin{array}{|l|l|l|l|l|l|}\hline \text { Gallons } & 5 & 10 & 20 & 25 & 30 \\\ \hline \text { Liters } & & & & & \\\\\hline\end{array}$$

Short answer

The mathematical model that relates gallons to liters is \( L = 53/14 * G \). Using this model, the completed table is: 5 gallons = \( 53/14 * 5 \) liters, 10 gallons = \( 53/14 * 10 \) liters, 20 gallons = \( 53/14 * 20 \) liters, 25 gallons = \( 53/14 * 25 \) liters, and 30 gallons = \( 53/14 * 30 \) liters.

Step-by-step solution

Identify the Conversion Rate

Given in the problem, 14 gallons of gasoline equates to 53 liters. This can be considered as the conversion rate. Thus, the ratio of gallons to liters is 14/53 or 1 gallon = 53/14 liters.

Develop the Mathematical Model

The mathematical model relating gallons to liters can be written as a function \( L = f(G) \) where \( L \) stands for Liters and \( G \) for Gallons. Given the conversion rate, the relationship can be formulated as \( L = 53/14 * G \).

Complete the Table

Using the equation \( L = 53/14 * G \), the conversion for each gallon amount can be calculated. Thus, for 5 Gallons, \( L = 53/14 * 5 \), for 10 Gallons, \( L = 53/14 * 10 \), for 20 Gallons, \( L = 53/14 * 20 \), for 25 Gallons, \( L = 53/14 * 25 \), and for 30 Gallons, \( L = 53/14 * 30 \). After calculating these, the corresponding liters can be filled in the given table.

Key concepts

Conversion Rate

When we talk about conversion rates in mathematics, especially in the context of unit conversion, we are referring to the ratio used to convert between two different units of measurement. In our exercise, we are looking at converting gallons to liters. This is a common real-world application because different countries use different systems of measurement for volume. Knowing how to switch from one to the other is an essential skill.

In this scenario, 14 gallons of gasoline correspond to 53 liters. This is our conversion rate. To find out how much one gallon is in liters, you simply divide 53 by 14. This gives us approximately 3.7857 liters per gallon.

A quick way to make conversions easier is to remember some key equivalences like this one:

• 1 gallon is approximately 3.7857 liters

This conversion rate allows us to set up calculations easily. Always verify the equivalence if precision is crucial—sometimes even small errors in conversion rate can lead to significant discrepancies when large quantities are involved.

Mathematical Modeling

Mathematical modeling is the process of creating a mathematical representation of a real-world situation. In this context, modeling is particularly useful for converting between units, because it allows us to apply a consistent method to make these conversions.

In our exercise, we established a model that relates gallons to liters using the conversion rate we identified. The function we formulated is given by \(L = f(G)\), which leads us to the equation \(L = \frac{53}{14} \times G\). Here, \(L\) represents liters and \(G\) represents gallons.

Using a mathematical model helps us:

• Ensure accuracy in our calculations
• Make predictions or fill in missing data easily
• Provide a clear framework that can be universally applied

This model is particularly powerful because it predicts the liters for any given number of gallons without needing further information.

Functions

Functions in mathematics are like a machine where you input something, it processes the input, and then gives you an output based on the rule it follows. Functions are important because they allow us to define and solve problems involving relationships between quantities.

In our scenario, the function \(L = \frac{53}{14} \times G\) is a simple linear function where you input the number of gallons, \(G\), and it outputs the equivalent number of liters, \(L\). Functions like these help us compute values quickly and handle repetitive conversions without needing to manually multiply the conversion rate every time.

The structure of functions facilitates ease of use:

• They provide a set formula to use repeatedly
• They illustrate how changing one variable affects another
• They allow for easy graphing and visualization of relationships

By understanding functions, you can quickly adapt to relate different quantities in a variety of contexts, extending beyond just gallons and liters.