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: Understand the Problem

We need to calculate the expected cost of a car in 2004 that originally cost $20,000 in 1999, given an annual inflation rate of 2.5%.

Step 2: Identify the Formula

We will use the formula for compound interest to calculate the expected cost: \[ \text{Final Price} = \text{Initial Price} \times (1 + \text{Inflation Rate})^n \]where \( n \) is the number of years.

Step 3: Plug Values into the Formula

Plug in the given values: - Initial Price = $20,000- Inflation Rate = 2.5% or 0.025- Number of years, \( n = 2004 - 1999 = 5 \)\[ \text{Final Price} = 20000 \times (1 + 0.025)^5 \]

Step 4: Perform the Calculation

Calculate the growth factor first: \[ 1 + 0.025 = 1.025 \]Then raise this growth factor to the power of 5:\[ 1.025^5 \approx 1.1314 \]Finally, multiply by the initial price:\[ 20000 \times 1.1314 \approx 22628 \]

Step 5: Interpret the Result

The expected cost of the car in 2004 would be approximately $22,628.

Key Concepts

Understanding the Inflation Rate

Inflation is the rate at which the general level of prices for goods and services rises, and subsequently, purchasing power falls. Central banks attempt to limit inflation—and avoid deflation—in order to keep the economy running smoothly. The inflation rate is expressed as a percentage. In our example, the inflation rate is given as 2.5% per year.

When prices inflate at 2.5%, it means that, on average, items like groceries, clothes, and also cars become 2.5% more expensive each year. Understanding this concept is crucial when calculating future values. It helps in estimating whether money saved today will have enough value to purchase the same goods in the future.

Thus, keeping track of inflation is essential in financial planning because it impacts various aspects of personal finance, from savings to investments.

Annual Percentage Increase

The annual percentage increase is the yearly rate at which a price grows due to compounding of previous years' growth. It is a key concept in financial mathematics, particularly when calculating compound interest. In simple terms, it's the percentage by which a price or value increases annually over a period.

In our exercise, the car's price subject to a 2.5% annual increase reflects this concept. Over the 5 years from 1999 to 2004, the price does not just increase by a straight 2.5% each year, but it increases on the already increased amount year over year. This is why the calculation involves raising the growth rate (1.025 in this case) to the power of the number of years ( = 5), highlighting how compounding accumulates over time.

The result of this calculation shows us how a modest annual growth rate compounds to a substantial price increase over multiple years.

Financial Mathematics in Real Life Applications

Financial mathematics involves using mathematical techniques to solve financial problems. It combines elements of algebra, calculus, and statistics to help us make sound financial decisions. This branch of mathematics is heavily applied in finance to model markets, manage risks, and evaluate investments.

In personal finance, understanding financial mathematics enables individuals to better understand how their money grows over time, including calculating loan interest or the future value of investments. The exercise we've explored above illustrates how such calculations can predict future costs through the compounding effect.

By appreciating these calculations, individuals can make informed decisions about savings and expenditures. For example, understanding how much inflation impacts purchasing power over time may encourage more strategic saving strategies or investment planning to preserve and grow wealth effectively.