Problem 94 - A Graphical Approach to College Algebra

Question

Converting Units of Measure Write a formula for a function \(f\) that converts \(x\) gallons to quarts. How many quarts are there in 19 gallons?

Step-by-Step Solution

Answer

There are 76 quarts in 19 gallons.

Step 1: Understand the Conversion Factor

The conversion factor between gallons and quarts is that 1 gallon is equal to 4 quarts. This means that to convert gallons to quarts, you multiply the number of gallons by 4.

Step 2: Formulate the Function

To find a formula for a function \(f\) that converts \(x\) gallons to quarts, use the conversion factor. The function is:\[ f(x) = 4x \]This means that the function \(f\) takes the number of gallons \(x\) and multiplies it by 4 to get the number of quarts.

Step 3: Apply the Function to the Given Problem

Apply the function \(f(x) = 4x\) to the given problem, where \(x = 19\) gallons. Substitute \(19\) into the function:\[ f(19) = 4 \times 19 \]

Step 4: Calculate the Result

Perform the calculation from the previous step:\[ f(19) = 4 \times 19 = 76 \]This means that 19 gallons is equivalent to 76 quarts.

Key Concepts

Gallons to Quarts

Converting between gallons and quarts is a practical skill when dealing with liquid volumes, especially in contexts like cooking or measuring fuel. Understanding that there are 4 quarts in a gallon is fundamental. When asked to convert any given number of gallons into quarts, simply multiply the number of gallons by 4.

This relationship is straightforward:

• 1 gallon = 4 quarts
• 2 gallons = 8 quarts
• 3 gallons = 12 quarts, and so on.

By keeping this simple multiplication in mind, you'll be able to quickly and accurately convert gallons into quarts without much hassle.

Conversion Factor

A conversion factor is a numerical factor used to multiply or divide a quantity when converting from one unit to another. In our case, the conversion factor from gallons to quarts is 4. This value stems from the definition that there are exactly 4 quarts in 1 gallon, providing us a means to transform gallons into quarts.

You use this factor to scale the original unit ( gallons) into the desired unit (quarts). For any number of gallons, simply multiply by this conversion factor (4) to obtain the equivalent number of quarts. This method assures an accurate conversion every time.

Function Formulation

Formulating a function for conversions is a useful exercise, as it allows us to generalize the conversion process. In this scenario, we need a function that takes the number of gallons, represented by a variable, and outputs the equivalent number of quarts.

Using the conversion factor, we create a function:

• Let the function be denoted as \( f(x) \).
• The input \( x \) is the number of gallons.
• The function is defined as \( f(x) = 4x \).

This indicates that any input value \( x \) (the number of gallons) is multiplied by 4 to convert it to quarts.

Mathematical Calculation

The final step in this conversion process involves applying our formulated function to a specific problem and performing the necessary arithmetic. To find out how many quarts are in 19 gallons, we use the function \( f(x) = 4x \) and substitute 19 for \( x \).

This calculation looks as follows:

• Substitute 19 in place of \( x \): \( f(19) = 4 \times 19 \)
• Multiply: \( 4 \times 19 = 76 \)

Thus, we calculate that 19 gallons is equal to 76 quarts. This shows how mathematical calculations help in translating a function into numerical results using straightforward arithmetic.