Problem 28 - Calculus Early Transcendentals: Pearson New International Edition

Question

Inflation between 1999 and 2004 ran at about \(2.5 \%\) per year. On this basis, what would you expect a car that would have cost \(\$ 20,000\) in 1999 to cost in \(2004 ?\)

Step-by-Step Solution

Answer

The car would cost approximately $22,628 in 2004.

Step 1: Understanding the Problem

We need to calculate the price of a car in 2004, given that the price in 1999 was $20,000 and inflation is 2.5% per year.

Step 2: Formula for Compound Interest

The formula to calculate the future value with annual inflation is: \( FV = PV \times (1 + r)^n \), where \( FV \) is the future value, \( PV \) is the present value, \( r \) is the annual rate of increase, and \( n \) is the number of years.

Step 3: Identify Variables

Here, \( PV = \$20,000 \), \( r = 0.025 \), and \( n = 5 \) years (from 1999 to 2004).

Step 4: Substitute Values into the Formula

Substituting the values into the formula: \[ FV = 20000 \times (1 + 0.025)^5 \]

Step 5: Calculate the Result

Calculate \( 1 + 0.025 = 1.025 \), then \( 1.025^5 \). Next, multiply the result by \( 20,000 \).

Step 6: Final Calculation

Calculate \( 1.025^5 \approx 1.1314 \) and then \( 20000 \times 1.1314 = \$22,628 \).

Key Concepts

Understanding Compound Interest

Compound interest is a fundamental mathematical concept that describes how an investment or cost grows over time by earning interest on both the initial principal and the accumulated interest from previous periods. In simpler terms, it's like earning interest on interest.
In the context of inflation, compound interest allows us to view costs or prices growing over time due to a persistent rate of increase. It mirrors how real-world prices rise due to economic factors.

• The formula for compound interest is: \( A = P \times (1 + r)^n \)
• Where \( A \) is the amount of money accumulated after n years, including interest.
• \( P \) is the principal amount (initial cost).
• \( r \) is the annual interest rate (or inflation rate in our case).
• \( n \) is the number of years the money is invested or borrowed.

In this exercise, we use this concept to understand how the price of a car could increase with inflation over a five-year period.

Inflation Calculation Over Time

Inflation is a measure of how prices for goods and services rise, indicating a decrease in purchasing power of money. When we look at inflation over a period of years, like in the exercise from 1999 to 2004, we use the concept of compound interest to understand its cumulative effect.
Inflation impacts the cost of goods each year by a certain percentage; here, it's given as 2.5%. Calculating inflation over a period means adjusting the initial price not just once, but compounding it over the time span.
Consider this approach for inflation calculation:

• Start with an initial price, known as the present value.
• Apply the rate of inflation repeatedly over the number of years (compounding).
• Use the compound interest formula to find the new price at the end of the period.

This presents the future value of an item, showing how much it will cost in the future after accounting for inflation.

Future Value Formula Application

The Future Value (FV) formula is crucial in predicting the value of an asset or item after inflation or interest has been applied over time. It's extensively used in finance to forecast future costs or profits.
In our exercise, applying the future value formula enabled us to find the projected cost of the car in 2004, starting from its 1999 price.
Here's how it works:

• The formula is \( FV = PV \times (1 + r)^n \), where:
• \( PV \) is the present value or initial cost of the item.
• \( r \) is the rate of increase per period (annual inflation rate here).
• \( n \) is the number of periods (years) the money is subject to growth.

By substituting the given values \( 20,000 \), \( 0.025 \), and \( 5 \) years into the formula, we calculated the future cost to be approximately \( 22,628 \). This represents a comprehensive view of the item's price trajectory, factoring in continuous inflation over the given years.