Problem 35 - College Algebra and Calculus: An Applied Approach

Question

Suppose that the annual rate of inflation averages \(4 \%\) over the next 10 years. With this rate of inflation, the approximate cost \(C\) of goods or services during any year in that decade will be given by \(C(t)=P(1.04)^{r}, \quad 0 \leq t \leq 10\) where \(t\) is time in years and \(P\) is the present cost. If the price of an oil change for your car is presently \(\$ 24.95\), estimate the price 10 years from now.

Step-by-Step Solution

Answer

The price of an oil change 10 years from now will be approximately \$36.95

Step 1: Identify the given values

In the problem, the present cost of the oil change \(P\) is \(\$24.95\), the annual rate of inflation \(r\) is \(4\%\) or \(0.04\) in decimal form and the time \(t\) is 10 years.

Step 2: Use the compound interest formula

The general formula for compound interest is \(C(t) = P(1 + r)^t\). Substituting the given values into the formula we get \(C(10) = \$24.95(1 + 0.04)^{10}\)

Step 3: Calculate the final cost

Applying the mathematics, this equates to \(C(10) = \$24.95 \times (1.04)^{10}\). Through calculation, the price of the oil change 10 years from now will be \$36.95 approximately.

Key Concepts

Compound Interest

Compound interest is a key financial concept used to calculate the future value of an investment or cost after a certain period. Unlike simple interest, where interest is calculated on the initial principal only, compound interest accumulates on both the initial principal and the interest that has been added to it over time. This means that not only is the initial amount invested earning interest, but the interest from previous years also earns interest.

In many scenarios, including the calculation of inflation over time, the compound interest formula becomes invaluable. The basic formula that describes compound interest is:

• \[ C(t) = P(1 + r)^t \] Where:
  • \( C(t) \) is the future cost or value,
  • \( P \) is the present value or initial cost,
  • \( r \) is the annual interest rate (or inflation rate), and
  • \( t \) is the time in years.

This formula allows us to see how costs or investments grow exponentially over time when compounded annually. By using this approach, we are better able to understand and estimate expenses and savings goals.

Exponential Growth

Exponential growth is a core concept in understanding how quantities increase at a consistently high rate relative to their current size. When applied to financial concepts, such as inflation or investments, it illustrates how values grow larger over time as a result of compounding.

Exponential growth can be visualized using the compound interest formula \( C(t) = P(1 + r)^t \). Here, notice that as \( t \) increases, the expression \((1 + r)^t\) grows exponentially. This implies that the cost of goods, such as an oil change in the context of inflation, will rise more significantly as time progresses. This is why it is important for individuals to consider inflation rates and investment returns when planning financially for the future.

Understanding exponential growth aids individuals in projecting the future value of expenses or savings. This enables them to make informed decisions about budgeting and financial planning.

Future Value Calculation

Future Value Calculation refers to determining the amount of money or cost of an item after a specific period, accounting for factors like interest or inflation. Calculating future value helps assess the impact of changes in cost over time.

Using the formula \( C(t) = P(1 + r)^t \), we can estimate what a current expense will cost in the future by applying a consistent rate of increase. For instance, in the original exercise, we calculate the future expense of an oil change given the inflation rate over 10 years. This involves:

• Identifying current costs (\( P \)).


• Defining the rate of inflation or interest (\( r \)).


• Specifying the time period (\( t \)).

The formula then provides us with the future value, informing us how much more expensive the service will be due to inflation over the defined period. Essentially, future value calculation allows individuals to anticipate and plan for economic changes, safeguarding their financial stability.