Problem 106 - Calculus Volume 1

Question

[T] 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

a. \( P(t) = 13,888.89t + 250,000 \). b. The slope is \$13,888.89/year. c. \$458,333.35 after 15 years.

Step 1: Understanding the Linear Function

A linear function has the form \( P(t) = mt + c \), where \( m \) is the slope and \( c \) is the y-intercept. We know the house's price is \\(250,000 initially (\( t=0 \)) and \\)500,000 after 18 years (\( t=18 \)). This gives us two points: \((0, 250,000)\) and \((18, 500,000)\).

Step 2: Calculating the Slope

The slope \( m \) of the line can be found using the formula: \[ m = \frac{P(t_2) - P(t_1)}{t_2 - t_1} \]. Substituting the values: \( t_1 = 0, P(t_1) = 250,000 \), \( t_2 = 18, P(t_2) = 500,000 \), we get \[ m = \frac{500,000 - 250,000}{18 - 0} = \frac{250,000}{18} = 13,888.89 \]. Thus, the slope is \(13,888.89\).

Step 3: Finding the Linear Function

Using the slope \( m = 13,888.89 \) and the point \((0, 250,000)\), the linear function is \( P(t) = 13,888.89t + 250,000 \).

Step 4: Interpreting the Slope

The slope \( m = 13,888.89 \) indicates the house's price increases by \$13,888.89 each year after its purchase.

Step 5: Calculating the House Price After 15 Years

To find the price of the house after 15 years, substitute \( t=15 \) into the linear function: \[ P(15) = 13,888.89 \times 15 + 250,000 \]. Calculating this gives \( P(15) \approx 458,333.35 \).

Key Concepts

Slope Interpretation

The slope of a linear function is crucial for understanding how a function behaves over time. When we talk about the slope, we refer to the rate at which one variable changes relative to another. In the context of a linear function, it's denoted by the letter \( m \) and indicates how steep the line representing the function is.

In our exercise, the slope is \( 13,888.89 \), which tells us that the price of the house increases by \$13,888.89 each year. This is a vital insight because it allows us to understand the rate of increase of the house's value over time.

• If the slope were steeper, the house price would be increasing more rapidly.
• If the slope were less steep, the increase in the house price would be more gradual.

Thus, the slope provides a direct way to visualize the rate of increase without having to calculate the price year by year.

Linear Equations

Linear equations are a key element of algebra that describe relationships with constant rates of change. They can be represented by the equation \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. In our case, the linear equation for the house’s price is modeled as \(P(t) = 13,888.89t + 250,000\).

Here:

• \(P(t)\) represents the house price after \(t\) years.
• \(13,888.89t\) describes how much the price increases each year considering \(t\) years of ownership.
• \(250,000\) is the initial price of the house, the y-intercept when \(t = 0\).

This equation can be used to predict the house price at any given year \(t\). By substituting different values for \(t\), students can see how the house's price changes over time and gain insight into the function's behavior.

Function Modeling

Function modeling involves using mathematical equations to represent real-world scenarios. It is an essential skill in mathematics because it helps us understand and predict changes.

In this example, the linear function \(P(t) = 13,888.89t + 250,000\) models how the price of a house evolves over time. Function modeling allows us to:

• Visualize future financial scenarios, like predicting house prices in future years.
• Test assumptions, such as the rates of change in value.
• Make informed decisions based on pricing trends and patterns.

With this function, we predicted the house price after 15 years, calculated as \(P(15) = 13,888.89 \times 15 + 250,000\), which gives an estimated value of \$458,333.35. Function modeling plays a powerful role in planning and understanding financial growth over time.