Problem 106 - Calculus Volume 1

Question

A house purchased for $$\$ 250,000$$ is expected to be worth twice its purchase price in 18 years. a. Find a linear function that models the price \(P\) of the house versus the number of years \(t\) since the original purchase. b. Interpret the slope of the graph of \(P\). c. Find the price of the house 15 years from when it was originally purchased.

Step-by-Step Solution

Answer

The price of the house in 15 years is approximately \$458,333.35. The slope means the price increases by \$13,888.89 per year.

Step 1: Determine the Final Price

The house is expected to be worth twice its original purchase price in 18 years. Therefore, the final price is \(2 \times 250,000 = 500,000\).

Step 2: Identify the Linear Function Form

The linear function that models the price \(P\) versus the number of years \(t\) is given by \(P(t) = mt + b\), where \(m\) represents the slope and \(b\) represents the y-intercept.

Step 3: Determine the Y-intercept

When the house was initially purchased, \(t = 0\) and \(P(0) = 250,000\). This indicates that the y-intercept \(b = 250,000\).

Step 4: Calculate the Slope

To find the slope \(m\), use the formula \(m = \frac{\Delta y}{\Delta x}\). Here, \(\Delta y = 500,000 - 250,000\) and \(\Delta x = 18 - 0\). Therefore, \(m = \frac{250,000}{18}\).

Step 5: Simplify the Slope

Compute the value of the slope: \(m = \frac{250,000}{18} \approx 13,888.89\).

Step 6: Write the Linear Function

Substitute \(m\) and \(b\) back into the linear function form: \(P(t) = 13,888.89t + 250,000\).

Step 7: Interpret the Slope

The slope \(13,888.89\) signifies that the price of the house increases by approximately \$13,888.89 each year.

Step 8: Calculate the House Price at 15 Years

Substitute \(t = 15\) into the linear function: \(P(15) = 13,888.89 \times 15 + 250,000\).

Step 9: Perform the Calculation

Calculate \(P(15)\): \(P(15) = 13,888.89 \times 15 + 250,000 = 458,333.35\).

Key Concepts

Understanding Slope Interpretation

In the context of linear functions, the slope is a critical concept that translates to a rate of change. It shows how one quantity changes in relation to another.
In the house price modeling exercise, the slope tells us how the price of the house increases year by year. When we calculate the slope (\( m \)), we use the formula:

• \( m = \frac{\Delta y}{\Delta x} \)

Here, \( \Delta y \) is the change in the house price and \( \Delta x \) is the change in years.
For our exercise, \( m \) ended up being approximately 13,888.89.
This number has an important interpretation: the price of the house is predicted to increase by roughly \$13,888.89 each year.
So, in practical terms:

• Steeper slope = Faster increase in value
• Flatter slope = Slower increase in value

This helps homeowners, investors, and analysts understand the expected growth rate over time.

Importance of the Y-Intercept

The y-intercept in a linear equation provides a starting point for understanding values in time-dependent scenarios. In our house price modeling exercise, the y-intercept (\( b \)) indicates the initial price of the house at the point of purchase. In a linear function of the form \( P(t) = mt + b \), the y-intercept is \( b \). For our specific case, this y-intercept was determined to be \$250,000. Here's why the y-intercept is important:

• It shows the initial property value at zero (beginning year)
• Helps in constructing the overall equation for future predictions

In simpler terms, the y-intercept acts as the ground zero of the price timeline.Knowing this starting point allows anyone analyzing the data to see how the house value begins and measure growth from that initial price over the years.

House Price Modeling Made Simple

Modeling house prices with a linear function is an effective way to forecast future values. Linear equations represent scenarios where there is a constant rate of change – perfect for our house price scenario. The linear function we derived, \( P(t) = 13,888.89t + 250,000 \), neatly encapsulates the expected price growth. Here's why this model is useful:

• Easy to predict house prices for future years
• Simple mathematical form that many can understand

For example, to predict the house price 15 years after the purchase, substituting \( t = 15 \) into the equation will give us \( P(15) \approx 458,333.35 \).Using such linear models facilitates decision-making for:

• Home buyers planning long-term investments
• Real estate investors looking for reliable forecasts
• Policy makers evaluating housing market trends

Thus, this approach not only aids in making informed financial decisions but also simplifies complex valuation processes.