Problem 91 - A Graphical Approach to Precalculus with Limits

Question

If the sales tax rate is \(7.5 \%,\) write a function \(f\) that calculates the sales tax on a purchase of \(x\) dollars. What is the sales tax on a purchase of \(\$ 86 ?\)

Step-by-Step Solution

Answer

The sales tax on a $86 purchase is $6.45.

Step 1: Define the Sales Tax Rate as a Function

The sales tax rate is given as \(7.5\%\). To convert this percentage into a decimal for calculation, divide by 100. So, \(7.5\% \) becomes \(0.075\). Thus, the function \(f(x)\) that calculates sales tax is defined as the product of the tax rate and the purchase amount, \( x \). Hence, \( f(x) = 0.075 \times x \).

Step 2: Calculate Sales Tax for a Specific Purchase

To find the sales tax on a purchase of \(\$86\), substitute \(x = 86\) into the function \(f(x) = 0.075 \times x \). This gives \(f(86) = 0.075 \times 86\).

Step 3: Perform the Calculation

Calculate \(0.075 \times 86\). First, multiply \(86\) by \(0.075\) to find the sales tax amount: \[86 \times 0.075 = 6.45\] So, the sales tax for a purchase of \(\\(86\) is \(\\)6.45\).

Key Concepts

Function Definition

One of the fundamental concepts in mathematics is defining a function to represent a real-world scenario. A function acts like a mathematical machine where you input a number, and it processes this number to give you an output. In the case of our sales tax problem, we define a function to calculate the tax on any given purchase amount.

To define a sales tax calculation function:\[ f(x) = ext{Tax Rate} \times x \]
Here, \(x\) represents the purchase amount.

This function allows you to compute the sales tax by simply plugging in the cost of an item. If the tax rate changes, you can easily adjust the function definition by changing the multiplier, making it a versatile tool to consistently determine tax amounts no matter the purchase value or changing rates.

Percentage Conversion

Understanding percentages is crucial when dealing with calculations such as sales tax. A percentage reflects a number out of 100. For computations, we need to convert percentages into a decimal form.

Here's how to convert a percentage to a decimal, which is key for applying it to any calculation:

• Take the percentage number (like 7.5%) and divide it by 100.
• This results in a decimal (for example, 7.5 divided by 100 equals 0.075).

Using the decimal form, multiply by the purchase amount to find out the actual amount of sales tax. This conversion is necessary to apply mathematical operations, as percentages in their original form cannot be directly multiplied with amounts in typical calculations.

Mathematical Operations

Mathematical operations form the core of calculating quantities like sales tax. The primary operation involved here is multiplication. Once you have the decimal form of the tax rate, multiplying this with the purchase amount gives the sales tax due.

To illustrate, consider a taxable purchase of \(\$86\) with a tax rate of 7.5%:

• Convert 7.5% to a decimal: \(0.075\)
• Multiply the purchase price by the decimal: \(86 \times 0.075 = 6.45\)

The result, \(6.45\), is the additional cost incurred due to tax, which should be added to the purchase amount for the total cost. These operations are vital in making sure that calculations are accurate and help in managing finances efficiently.