Problem 70 - Algebra and Trigonometry with Analytic Geometry

Question

A proposed energy tax \(T\) on gasoline, which would affect the cost of driving a vehicle, is to be computed by multiplying the number \(x\) of gallons of gasoline that you buy by 125,000 (the number of BTUs per gallon of gasoline) and then multiplying the total BTUs by the tax \(-34.2\) cents per million BTUs. Find a linear function for \(T\) in terms of \(x\).

Step-by-Step Solution

Answer

The linear function for the energy tax is \(T(x) = -0.04275x\).

Step 1: Understand the problem

We need to express the energy tax \(T\) as a linear function of the number of gallons \(x\) of gasoline purchased.

Step 2: Calculate BTUs per gallon

Each gallon of gasoline generates 125,000 BTUs. Therefore, if you buy \(x\) gallons of gasoline, the total BTUs will be \(125,000x\).

Step 3: Convert the tax rate to dollars

The tax rate is given as \(-34.2\) cents per million BTUs. Since 1 dollar = 100 cents, this is equivalent to \(-0.342\) dollars per million BTUs.

Step 4: Calculate the tax on total BTUs

First, convert \(125,000x\) BTUs to million BTUs by dividing by 1,000,000: \[\text{Total Million BTUs} = \frac{125,000x}{1,000,000} = 0.125x.\] Now, calculate the tax \(T\) using the rate \(-0.342\) dollars/million BTUs: \[T = -0.342 \times 0.125x = -0.04275x.\]

Step 5: Write the linear function

The linear function for \(T\) in terms of \(x\) is \[T(x) = -0.04275x.\]

Key Concepts

Energy Tax Calculation

Calculating an energy tax involves a few straightforward steps that transform a consumption measurement into a monetary amount. In this specific case, the energy tax is meant to assess the cost associated with burning gasoline in a vehicle. To find this cost, we create a linear equation: a mathematical function that linearly relates one variable to another.

**Breaking Down the Calculation**

• The number of gallons of gasoline you purchase is denoted by the variable \(x\).
• Each gallon contributes 125,000 BTUs of energy.
• To determine the energy tax \(T\), you first calculate the total BTUs by multiplying the number of gallons \(x\) by 125,000 BTUs.
• The given tax rate is \(-34.2\) cents per million BTUs. This needs to be converted to dollars, resulting in \(-0.342\) dollars per million BTUs.

This plan allows you to easily convert the physical consumption of gasoline into a financial cost, helping understand how energy usage translates into expenses.

BTUs Conversion

BTUs, or British Thermal Units, are essential for measuring energy. They are particularly useful when calculating energy outputs from fuels like gasoline. In this exercise, you need to determine how many million BTUs arise from the gasoline purchased.

**Step-by-Step Conversion**

• For each gallon, gasoline produces 125,000 BTUs.
• Multiply this value by your number of gallons \(x\) to find the total BTUs.
• Convert these BTUs into "million BTUs" by dividing the total BTUs by 1,000,000: \( \text{Total Million BTUs} = \frac{125,000x}{1,000,000} = 0.125x \).

This conversion is crucial because energy taxes are often calculated per million BTUs. This keeps the numbers manageable and standardized, simplifying comparisons and applications.

Coefficient of x in Linear Functions

In the created linear function \(T(x) = -0.04275x\), the coefficient of \(x\) is \(-0.04275\). Understanding this coefficient is an important part of interpreting how the function behaves. It reveals the rate of change or how much the dependent variable \(T\) changes for every one-unit increase in the independent variable \(x\).

**Understanding the Coefficient**

• The coefficient represents the energy tax rate applied per gallon of gasoline.
• In monetary terms, the cost increases by \(-0.04275\) dollars (or approximates to a decrease, noting the negative sign), as gallons \(x\) increase by one.
• The negative value indicates the cost associated is recognized as an expense, depicting the tax you owe.

This knowledge helps translate mathematical functions into real-world financial implications, offering a clear insight into how gasoline consumption directly impacts expenses.