Problem 80 - Algebra and Trigonometry Real Mathematics, Real People

Question

Assume the annual rate of inflation is \(4 \%\) for the next 10 years. The approximate cost \(C\) of goods or services during these years is \(C(t)=P(1.04)^{t},\) where \(t\) is the time (in years) and \(P\) is the present cost. An oil change for your car presently costs \(\$ 26.88 .\) Use the following methods to approximate the cost 10 years from now. (a) Use a graphing utility to graph the function and then use the value feature. (b) Use the table feature of the graphing utility to find a numerical approximation. (c) Use a calculator to evaluate the cost function algebraically.

Step-by-Step Solution

Answer

All methods should end up approximately with the same result. Calculating the function \(C(t)=P(1.04)^{t}\) for \(t=10\) and \(P=26.88\) yields a value of approximately \$40.06. So, an oil change will cost roughly \$40.06 in 10 years given a constant annual inflation rate of 4%.

Step 1: Understand the Function

The inflation function given is \(C(t)=P(1.04)^{t}\), where \(C(t)\) is the future cost of goods or services, \(P\) is the present cost, \(t\) is the time in years, and 1.04 represents a 4% annual inflation rate. An oil change currently costs $26.88.

Step 2: Use Graphing Utility for Value

Graph the function \(C(t) = 26.88*(1.04^t)\) using a graphing calculator or tool. Then, use the value feature on the graphing tool to find the y-coordinate of the point on the graph where \(t=10\). The y-coordinate will represent the cost of the oil change 10 years from now.

Step 3: Use Table Feature for Numerical Approximation

Use the table feature of the graphing utility, populating the x-values (representing 't') with numbers from 0 to 10. Under \(t=10\)), there will be a corresponding y-value. This y-value is the cost of the oil change in 10 years.

Step 4: Calculate Algebraically

Plug the values of \(P=26.88\), \(t = 10\) and constant inflation into the given formula to calculate manually. You'll find \(C(10)= 26.88 * (1.04)^{10}\)

Key Concepts

Inflation Rate

When we talk about inflation, we're referring to the general increase in prices and the fall of purchasing power over time. For example, if the inflation rate is 4% annually, it means that, on average, prices for goods and services rise by 4% each year. So, something that costs \(100 this year would cost \)104 next year. This is important in economic planning, as it impacts everything from the cost of day-to-day items to larger long-term expenses, like an oil change for your car.

In this context, inflation affects how the future costs of services, like car maintenance, are calculated. The formula used here is an exponential function: \(C(t) = P(1.04)^t\). Here, \(1.04\) represents the 4% annual increase, applied over time \(t\) years. Understanding this helps you see not only the importance of accounting for inflation but also how it compounds over time.

Cost Function

A cost function is a mathematical expression that represents how much a particular product or service will cost over time, taking into account various factors like inflation. In our example, the cost function is written as \(C(t) = P(1.04)^t\).

• \(C(t)\) represents the cost at any given time \(t\) in the future.
• \(P\) is the present cost of the item, which is \$26.88 for an oil change.
• \(1.04\) reflects the annual inflation rate, compounded yearly.

This function is called exponential because the cost at time \(t\) is multiplied by the inflation rate raised to the power of \(t\). This type of growth means that costs can increase quickly over time, making it a powerful tool for forecasting expenses. Being able to compute such functions lets you prepare for financial needs effectively.

Future Value

The concept of future value helps predict how much something will be worth in the future, taking into account factors like interest or inflation. In this case, we're interested in the future value of an oil change, considering an average inflation rate.

The equation \(C(t) = 26.88 * (1.04)^t\) calculates this future cost. Let's break it down:

• \(t = 10\) represents 10 years into the future.
• \(26.88\) is the current cost.
• \(1.04^t\) shows how much this cost will grow annually due to inflation.

After calculating \(C(10) = 26.88 \times (1.04)^{10}\), you'll find the expected cost in ten years, giving you the future value of the oil change, factoring in inflation. Understanding future value is crucial for long-term financial planning, as it helps predict future costs and allows for saving and budgeting accordingly.