Problem 24 - Algebra and Trigonometry Real Mathematics, Real People

Question

When \(\$ 1\) is invested in an account over a 10-year period, the amount \(A\) in the account after \(t\) years is given by \(A=1+0.06 t \quad\) or \(\quad A=\left(1+\frac{0.055}{365}\right)^{365 t}\) depending on whether the account pays simple interest at \(6 \%\) or compound interest at \(5 \frac{1}{2} \%\) compounded daily. Use a graphing utility to graph each function in the same viewing window. Which grows at a greater rate?

Step-by-Step Solution

Answer

The comparison will show the graph for the compound interest increases more steeply. This indicates that when interest is compounded daily, it grows at a greater rate than simple interest.

Step 1: Graph the simple interest formula

The equation for the simple interest is \(A=1+0.06 t\). It is a linear function and you can graph it by selecting a range for \(t\) and calculating the corresponding \(A\) values. Use a graphing tool for this purpose.

Step 2: Graph the compound interest formula

The compound interest is represented by the equation \(A=\left(1+\frac{0.055}{365}\right)^{365 t}\). This formula is a bit more complex due to daily compounding. Like step 1, choose a range for \(t\) and calculate corresponding \(A\) using the formula. Plot this function on the same graph to compare them both.

Step 3: Compare the growth rates

Now that both functions are plotted on the same graph, we can visually inspect which one grows at a higher rate. The function which increases more steeply will represent the higher growth rate over the given period of time.

Key Concepts

Simple Interest

Simple interest is a straightforward way of calculating interest on a loan or investment. The formula for calculating simple interest is \( A = P(1 + rt) \) where:

• \( P \) represents the principal amount (the initial sum of money)
• \( r \) represents the annual interest rate (in decimal)
• \( t \) represents the time the money is invested for, in years

In the case of the given problem, the simple interest formula simplifies to \( A = 1 + 0.06t \) because the principal amount, \( P \) is \$1 and the annual interest rate, \( r \) is 6% or 0.06 in decimal form. This results in a linear function which means that the amount of interest earned each year is constant. Consequently, if you were to graph this function, you would see a straight line that slopes upwards, reflecting the steady increase of the account balance over time.

Compound Interest

Compound interest, on the other hand, is a bit more complex. It calculates interest not only on the initial sum but also on the accumulated interest of previous periods. The general formula for compound interest is \( A = P \times \bigg(1 + \frac{r}{n}\bigg)^{nt} \) where:

• \( P \) is the principal amount
• \( r \) is the annual interest rate (in decimal)
• \( t \) is the time the money is invested for, in years
• \( n \) is the number of times that interest is compounded per unit \( t \)

The problem specifies a daily compounding interest rate, which means \( n = 365 \) times a year. The yearly rate is 5.5%, or \( r = 0.055 \). Inserting these values into the compound interest formula, we get \( A = \bigg(1 + \frac{0.055}{365}\bigg)^{365t} \) This will produce an exponential function when graphed, showing a more rapid increase in account balance, particularly over the long term, because the interest earned itself earns interest.

Graphing Functions

Graphing functions is an essential tool in visualizing how different variables in an equation relate to one another. When graphing the simple and compound interest functions from our exercise, we are able to compare the growth of two different types of interest over time. For the linear simple interest function, graphing is straightforward as you plot the increase over time based on a constant rate. With compound interest, the exponential nature means plotting the amount at different intervals to show how the growth accelerates due to the interest on interest effect. A graphing calculator or software can handle both functions efficiently, plotting them on the same set of axes to allow for easy comparison. Through graphing, it becomes visually apparent which type of interest yields a greater amount over time and can significantly aid in understanding how different financial options might play out.

Exponential Growth

Exponential growth occurs when the growth rate of a mathematical function becomes increasingly rapid in relation to the growing total number or size. In finance, compound interest is a prime example of exponential growth because the accumulated interest is added to the principal amount, and then new interest is calculated on this larger balance during the next compounding period.

In our specific scenario, as the compound interest formula includes raising a base to the power of the product of time and the number of compounding periods per year, this represents exponential growth. This growth can be contrasted with the linear growth of simple interest, which adds a fixed amount of interest each year. Over time, the account with compound interest accumulates wealth much faster than the account with simple interest, showcasing the power of exponential growth in financial contexts.