- Identify initial parameters: \(P = \$193,750\), \(r = 0.0425\), and \(t = 4\).
- Calculate the annual appreciation: \(193,750 \times 0.0425 = 8,234.375\).
- Calculate total appreciation over 4 years: \(8,234.375 \times 4 = 32,937.5\).
- Determine final price: \(193,750 + 32,937.5 = 226,687.5\), which rounds to \(\boxed{\$226,688}\).
Explanation
- Data and Problem Analysis
The townhouse is initially priced at \(P = \$193,750\). It appreciates at a rate of \(r = 4.25\%\) per year, which is equivalent to \(r = 0.0425\) in decimal form. The timeframe for this appreciation is \(t = 4\) years. The goal is to calculate the final price using the straight-line method, where the appreciation is calculated based on the original price each year.
- Calculating Total Appreciation
The straight-line method for appreciation calculates the increase in value as a linear function of time. The formula for the total final value \(A\) is:
\[A = P + (P \times r \times t)\]
First, we find the appreciation amount for one year:
\[193,750 \times 0.0425 = 8,234.375\]
Then, we multiply this annual increase by the total number of years (4) to find the total appreciation over the 4-year period:
\[8,234.375 \times 4 = 32,937.5\]
- Final Calculation and Conclusion
To find the final price \(A\), we add the total appreciation to the initial price:
\[A = 193,750 + 32,937.5 = 226,687.5\]
Rounding this value to the nearest dollar, we get:
\[A \approx \$226,688\]
This matches option a.
Examples
The straight-line method is commonly used for estimating the future value of assets when the rate of growth is expected to remain constant relative to the initial purchase price. For example, if a homeowner buys a property and wants to forecast its value for tax planning or insurance purposes, they might assume a stable annual appreciation rate. This calculation helps in making informed financial decisions about long-term investments and understanding the impact of steady growth over time.