Problem 18 - College Algebra and Calculus: An Applied Approach

Question

An item that sells for $$\$ 145.99$$ has a sales tax of $$\$ 10.22$$ (a) Find a mathematical model that gives the amount of sales tax \(y\) in terms of the retail price \(x\). (b) Use the model to find the sales tax on a purchase that has a retail price of \(\$ 540.50\).

Step-by-Step Solution

Answer

The sales tax on a purchase that has a retail price of \$540.50 is \$37.83.

Step 1: Determine the Tax Rate

First, calculate the tax rate from the given 'Sales Tax' \$10.22 and 'Retail Price' \$145.99. That is, m = y / x = \$10.22 / \$145.99 = 0.07.

Step 2: Formulate the Mathematical Model

On basis of the calculated tax rate, formulate the mathematical model for sales tax. In this case, the model is y = 0.07x.

Step 3: Plug in the Given Value

Now, use this model to compute the sales tax for a purchase with a retail price of \$540.50. Substitute x with \$540.50 in the model: y = 0.07 * \$540.50

Step 4: Compute the Sales Tax

Calculate the value of y which equals the sales tax on the given retail price, that is y = \$37.83.

Key Concepts

Mathematical Model

A mathematical model is like a magic formula. It helps us relate different quantities in a way that we can predict one using the other. In the context of sales tax, our model will tell us how much tax to expect based on the retail price of an item. For this exercise, we are using an equation to represent the relationship between the sales tax and the retail price. This equation is an example of building a mathematical model: \[ y = mx \] where

• \( y \) represents the sales tax,
• \( m \) is the tax rate, and
• \( x \) is the retail price.

The model helps you find the tax on any item, as long as you know the retail price.

Tax Rate

The tax rate is a percentage that tells us how much tax will be charged on an item's purchase price. Think of it as a portion of the price you pay, which adds to government revenue. In our case, the exercise gives us a sales tax of \( \\(10.22 \) on an item priced at \( \\)145.99 \). When calculating the tax rate:

• First, take the amount of sales tax, \( y \), which is \( \\(10.22 \).
• Then, divide it by the retail price, \( x \), \( \\)145.99 \).
• We get \( m = \frac{10.22}{145.99} \approx 0.07 \).

This means the tax rate is 7%. This percentage is crucial for calculating the tax on a new retail price.

Retail Price

Retail price refers to the cost at which you purchase an item before any tax is added. It's the price tag you see when you're shopping. In the exercise, the retail price is \( \\(145.99 \) for the initial example and \( \\)540.50 \) for the additional calculation. Understanding this price is crucial because:

• The tax is a percentage of the retail price.
• Without knowing the retail price, you can't calculate the exact amount of sales tax.

Retail price is the starting point for all further calculations, so make sure to get that number right before moving forward with any computations.

Algebraic Expression

An algebraic expression involves numbers, variables, and operators (like addition and multiplication) to show a mathematical relationship. In our sales tax calculation, the expression is given by:\[ y = 0.07x \]Here, \( y \) stands for the sales tax and \( x \) is the retail price. The number \( 0.07 \) is our tax rate, which we derived in previous steps. This expression allows you to easily compute the tax by just plugging in the given retail price. For example, if the retail price is \( \$540.50 \), the expression becomes: \[ y = 0.07 \times 540.50 \] This will compute the sales tax automatically, making algebra a handy tool for such calculations. With algebraic expressions, you can turn the complex process of sales tax calculation into a simple and efficient step.