Problem 47 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

PROPERTY TAX Khalil is trying to decide between two competing property tax propositions. With Proposition A, he will pay \(\$ 100\) plus \(8 \%\) of the assessed value of his home, while Proposition B requires a payment of \(\$ 1,900\) plus \(2 \%\) of the assessed value. Assuming Khalil's only consideration is to minimize his tax payment,develop a criterion based on the assessed value \(V\) of his home for deciding between the propositions.

Step-by-Step Solution

Answer

Choose Proposition A if the assessed value is less than \(30,000. Choose Proposition B if the assessed value is \)30,000 or more.

Step 1: Define the Tax Payments for Each Proposition

First, express the total tax payment for each proposition as a function of the assessed value of the home, denoted by lowercase v.For Proposition A, the tax payment is given by: a(v) = 100 + 0.08vFor Proposition B, the tax payment is given by: b(v) = 1900 + 0.02v

Step 2: Set Up an Inequality to Compare the Propositions

Next, determine when the tax payment for Proposition A is less than the tax payment for Proposition B by setting up the inequality: 100 + 0.08v < 1900 + 0.02v

Step 3: Solve the Inequality

Isolate the variable v on one side of the inequality:Subtract 0.02v from both sides:100 + 0.08v - 0.02v < 1900 Simplify:100 + 0.06v < 1900Subtract 100 from both sides:0.06v < 1800Divide by 0.06:v < 30000

Step 4: Interpret the Result

The inequality v < 30000 means that Proposition A is preferable when the assessed value of the home, v, is less than \(30,000. Conversely, if the assessed value is \)30,000 or greater, Proposition B becomes the better option.

Key Concepts

assessed value

The assessed value of a property is a significant factor in determining property taxes. Assessed value is an estimate of a property's worth used by tax authorities to calculate the amount of taxes due. The assessment considers various factors like the property's location, size, and market conditions. Khalil needs to understand the assessed value of his home to decide which tax proposition minimizes his payments. For the problems related to property tax, let’s denote this assessed value by the variable \(v\). It's crucial to determine the accurate assessed value, as it directly influences the tax calculations.

tax payment functions

Tax payment functions represent the total tax owed as mathematical expressions. In our problem, we have two propositions with different structures.
For Proposition A, Khalil’s total tax payment is defined as: \[ a(v) = 100 + 0.08v \] Here, \(100\) is a fixed amount, and \(0.08v\) accounts for 8% of the assessed value \(v\).
For Proposition B, the tax payment function is: \[ b(v) = 1900 + 0.02v \] Here, \(1900\) is the fixed amount, and \(0.02v\) accounts for 2% of the assessed value \(v\). Understanding these functions allows us to compare the tax liabilities under each proposition for different property values.

inequality solving

To determine which proposition offers a lower tax payment, we need to compare the two tax payment functions. This is done using inequalities. We want to find when Proposition A's payment is less than Proposition B's, as this indicates a lower tax.
The inequality to solve is: \[ 100 + 0.08v < 1900 + 0.02v \] To solve, we:

• Subtract \(0.02v\) from both sides: \[ 100 + 0.06v < 1900 \]
• Subtract \(100\) from both sides: \[ 0.06v < 1800 \]
• Divide by \(0.06\): \[ v < 30000 \]

This solution shows that Proposition A is cheaper when the assessed value \(v\) is less than \(30,000\).

comparison of propositions

Comparison of propositions involves interpreting the results to make an informed decision. From our inequality, \[ v < 30000 \]
we conclude the following:

• If the assessed value is less than \(30,000\), Proposition A results in lower taxes.
• If the assessed value is \(30,000\) or higher, Proposition B is more economical.

Therefore, knowing the assessed value of his home can guide Khalil to pick the proposition that minimizes his tax burden. Such comparisons help in understanding different tax schemes and making financially sound decisions.