Problem 17 - College Algebra and Calculus: An Applied Approach

Question

Your property tax is based on the assessed value of your property. (The assessed value is often lower than the actual value of the property.) A house that has an assessed value of $$\$ 150,000$$ has a property tax of $$\$ 5520$$. (a) Find a mathematical model that gives the amount of property tax \(y\) in terms of the assessed value \(x\) of the property. (b) Use the model to find the property tax on a house that has an assessed value of \(\$ 185,000\).

Step-by-Step Solution

Answer

The mathematical model is \(y = 0.0368x\), the tax on a property valued at $185,000 would be \$6808.

Step 1: Understand the Problem

The property tax \(y\) is directly proportional to the assessed value of the property \(x\). The constant of proportionality is unknown, but using the data given, we can denote it as \(k\). So, we can express the relationship as \(y = kx\). Our aim is to find \(k\) from the data provided.

Step 2: Substitute the Known Values

Using the known values of \(y = \$ 5520\) and \(x = \$ 150,000\), substitute in the equation \(y = kx\). This gives \$5520 = k * \$150,000.

Step 3: Solve for 'k'

Solving for \(k\) in the equation \$5520 = k * \$150,000 we get \(k = \frac{5520}{150000} = 0.0368\). This means, for every dollar of assessed value, the property tax increases by 0.0368 dollar.

Step 4: Formulate the Model

Now that we have found \(k\), we can write the mathematical model for the property tax as \(y = 0.0368x\). This is the answer to part (a) of the exercise.

Step 5: Apply the Model to a New Scenario

Now we apply the model to find the tax on a property assessed at $185,000. So, substituting \(x = \$185000\) into the model \(y = 0.0368x\), we get \(y = 0.0368 * \$185,000 = \$6808\). This is the answer to part (b) of the exercise, the property tax on a house valued at $185,000 would be $6808.

Key Concepts

Proportional Relationships

Understanding proportional relationships is essential when solving problems involving ratios and direct proportionality. In the context of property tax, the tax amount \(y\) is directly proportional to the assessed value of the property \(x\). This means that as the assessed value increases, the tax increases at a consistent rate.

Imagine a simple scenario: if a property valued at \$100 incurs a tax of \$5, then a property valued at \$200, under a proportional tax system, will incur a tax of \$10. The tax rate remains the same regardless of the property value, illustrating a proportional relationship. The constant of proportionality \(k\) here ensures that the same ratio applies to any assessed value, which means \(y\) can be predicted using the formula \(y = kx\).

Solving for Constants

Solving for constants involves finding the unknown values that make a mathematical expression or equation work. In our property tax scenario, we know the relationship between assessed property value and tax is proportional, but we don't initially know the constant of proportionality, \(k\).

To find \(k\), we use known values as a reference point. From the exercise, we have a house assessed at \$150,000 with a corresponding property tax of \$5,520. By substituting these known values into the equation \(y = kx\), we can solve for \(k\). It's like having a puzzle with one missing piece: once you find the value of that piece (\(k\) in this case), everything else falls into place. Once \(k\) is known, the model becomes a powerful tool for calculating property taxes at different assessed values.

Applied Mathematics

Applied mathematics is the application of mathematical methods by different fields such as science, engineering, business, and industry. In the context of our problem, applied mathematics allows us to create a mathematical model for real-world situations, like calculating property tax based on the assessed value of real estate.

This practical use of mathematics in our daily lives is what makes it such a vital tool. Part (a) of the exercise demonstrates the creation of a mathematical model, \(y = 0.0368x\), which is derived from understanding proportional relationships and solving for the constant of proportionality. Part (b) then applies this model to find the tax on a property with a different assessed value, showcasing how mathematics helps us solve tangible problems. Through applied mathematics, we simplify complex situations and answer pertinent questions that arise in various fields, including finance and economics.