Problem 76 - Precalculus Mathematics for Calculus

Question

Price of a House The median price of a house in Orange County increases by about \(6 \%\) per year. In 2002 the median price was \(\$ 240,000 .\) Let \(P_{n}\) be the median price \(n\) years after 2002 (a) Find a formula for the sequence \(P_{n}\) (b) Find the expected median price in 2010 .

Step-by-Step Solution

Answer

The formula is \( P_{n} = 240,000 \times (1.06)^n \). In 2010, the price is approximately $382,524.

Step 1: Understanding the Problem

We know the median house price increases by 6% each year and the initial price in 2002 was $240,000. We need to find a formula to represent the price in any subsequent year and a specific price for 2010.

Step 2: Setting Up the Formula for Price Growth

Since the price increases by 6% each year, you can express year-to-year growth as a multiplication by a factor of 1.06. Therefore, the price for any year can be represented by the formula: \[ P_{n} = 240,000 \times (1.06)^n \] where \( n \) is the number of years after 2002.

Step 3: Determine the Years Since 2002 Until 2010

Calculate how many years have passed from 2002 to 2010. Subtract 2002 from 2010: \[ n = 2010 - 2002 = 8 \] Thus, \( n = 8 \) years.

Step 4: Calculate the Median Price in 2010

Now substitute \( n = 8 \) into the formula: \[ P_{8} = 240,000 \times (1.06)^8 \] Calculate the value using a calculator. \[ P_{8} \approx 240,000 \times 1.59385 \approx 382,524 \raisebox{0.5ex}{-}\raisebox{0.5ex}{-} \]

Key Concepts

sequence formula

A sequence formula is a mathematical way to describe a pattern or a series of numbers, often used to predict future values based on initial conditions. In this exercise, we use a sequence formula to model the growth of median house prices over time. Given that - Starting price in 2002 is \( \$240,000 \),- The annual growth rate is \( 6\% \),We can derive the sequence formula by considering that each year's price is \( 1.06 \) times the previous year's price. The sequence formula for the median house price \( P_n \) is:\[ P_n = 240,000 \times (1.06)^n \] where \( n \) represents the number of years after 2002.
Just to break it down, this formula tells us that the price each year will be \( 240,000 \) multiplied by \( 1.06 \) raised to the power of \( n \), showing how prices exponentially grow year over year. This approach is very useful when handling scenarios with consistent growth patterns, making it easy to forecast near-term future prices.

percentage increase

Percentage increase is a measure that determines how much a quantity grows, relative to its original amount. In this context, we are dealing with a 6% annual increase in the median house price.

• The percentage increase is expressed as a multiplier, in this case, \( 1.06 \) which means a 6% addition for every new year.
• To calculate this, multiply the original figure by 1 plus the percentage growth rate represented as a decimal.

So each year, the new price becomes 6% higher than the previous year:
If you start with \( 240,000 \), then after one year:\[ 240,000 \times 1.06 = 254,400 \]
After 2 years:\[ 254,400 \times 1.06 = 269,664 \]
This process continues annually, each time multiplying last year's price by \( 1.06 \). This explains the compounding effect of the percentage increase, which adds more to the price each subsequent year.

median house price

The median house price represents the middle point of real estate prices in a given market. The median is used because it minimizes the effect of anomalies, such as extremely high or low house prices, giving a more reliable indication of typical market costs.In Orange County, the median house price in 2002 was \( \\(240,000 \). This exercise helps us understand both how this initial value can be used to calculate future prices, and how it influences our understanding of market trends.

• The median provides insights into the affordability and standard of living in an area.
• A consistent annual increase in the median house price highlights upward market trends, often reflecting increased demand and growth in the region.

When using the sequence formula, we can predict the anticipated future median price for any given year. For instance, we found that in 2010, roughly 8 years post-2002, the median price was expected to rise to \( \\)382,524 \). This showcases how the past median price, alongside consistent percentage increases, can project future market positions.