Problem 34 - Algebra Form and Function

Question

Find a formula for \(g\) by scaling the output of \(f\). Let \(f(t)\) give the number of liters of fuel oil burned in \(t\) hours, and \(g(t)\) the number of gallons burned. Use the fact that 1 gal equals 3.785 liters.

Step-by-Step Solution

Answer

Answer: The formula to convert f(t) to g(t) is g(t) = f(t) / 3.785.

Step 1: Write the conversion factor as an equation

Using the conversion factor provided, we can write an equation relating liters and gallons: 1 gal = 3.785 L Since we are given \(f(t)\) is in liters and \(g(t)\) is in gallons, we want to find an equation that expresses \(f(t)\) in terms of \(g(t)\).

Step 2: Convert the equation into a formula

To find a relationship between \(f(t)\) and \(g(t)\), we can start by expressing 1 gallon in terms of liters: 1 gal = 3.785 L Now, let's rewrite the equation above in terms of \(f(t)\) and \(g(t)\): \(g(t)\) gal = \(f(t)\) L

Step 3: Solve for \(g(t)\)

We want to find a formula for \(g(t)\) in terms of \(f(t)\). To do this, we will divide both sides of the equation by the conversion factor: \(g(t)\) gal = \(\frac{f(t) L}{3.785 \frac{L}{gal}}\) Now simplify the equation: \(g(t) = \frac{f(t)}{3.785}\) Now the formula for \(g(t)\) has been found in terms of \(f(t)\): \(g(t) = \frac{f(t)}{3.785}\) This is the formula for converting the number of liters of fuel oil burned (\(f(t)\)) into the number of gallons burned (\(g(t)\)).

Key Concepts

Scaling Output

When we talk about scaling output, we mean adjusting a given value by a particular factor. In the context of functions, this often means changing one measurement into another, making it easier to understand or use. For example, if you have a function that provides output in terms of one unit of measure and you need it in another, you'll need to "scale" the output appropriately. This involves using a conversion factor to modify each output value to the desired unit.

In the original exercise, we have two functions, where one describes the fuel consumption in liters and the other in gallons. To move from one measurement to another, we apply a conversion factor. This conversion adjusts the 'scale' of output from the units in liters to gallons, providing a direct comparison between the fuel consumption represented by each function.

Fuel Consumption

Fuel consumption refers to the amount of fuel used over a specific period, often measured in terms of volume. In the context of our exercise, fuel consumption is initially measured in liters through the function \( f(t) \), which defines how many liters are consumed over time "t," expressed in hours.

To monitor and reduce expenses or emissions, understanding consumption in different units — like gallons — is important. It can be particularly helpful if gallons are the more commonly used measure in certain regions or industries. Functions like \( g(t) \) help recalibrate usage by providing a different perspective on the same consumption patterns, allowing for informed decision-making and planning concerning fuel use.

Conversion Factor

Conversion factors are crucial tools used to convert measures from one unit to another. They are essentially multipliers that bridge differences between various units of measurement.

In this situation, the conversion factor between liters and gallons is provided: 1 gallon equals 3.785 liters. This means each liter can be mathematically transformed into a gallon by dividing by 3.785. This is clear in step three of the original solution, where both sides of the equation are divided by this factor to isolate \( g(t) \).

When utilizing conversion factors, remember:

• These factors simplify comparisons and calculations across different measurement systems.
• They ensure accuracy by maintaining consistent ratios between the units.
• Always verify the context and ensure the factor is correct for your specific calculation.

By comprehending and applying conversion factors correctly, solving problems related to unit conversion becomes straightforward and effective.