Problem 77 - Algebra and Trigonometry Real Mathematics, Real People

Question

There are three options for investing \(\$ 500 .\) The first earns \(7 \%\) compounded annually, the second earns \(7 \%\) compounded quarterly, and the third earns \(7 \%\) compounded continuously. (a) Find equations that model the growth of each investment and use a graphing utility to graph each model in the same viewing window over a 20-year period. (b) Use the graph from part (a) to determine which investment yields the highest return after 20 years. What are the differences in earnings among the three investments?

Step-by-Step Solution

Answer

The formulas representing the three investment schemes are \(A = 500(1+0.07)^t\), \(A = 500(1+0.07/4)^{4*t}\) and \(A = 500e^{0.07t}\) for annually, quarterly and continuously compounded interest respectively. Determine the highest return by comparing the graphs at \(t = 20\). The differences in earnings can be calculated by subtracting each of the lesser returns from the highest one.

Step 1: Define the formulas

The formulas for the three methods of compounding are as follows: - Annually: \(A = P (1 + r/n)^{nt}\), where \(n=1\)- Quarterly: \(A = P (1 + r/n)^{nt}\), where \(n=4\)- Continuously: \(A = P e^{rt}\)In our case, \(P = \$500\), \(r = 0.07\) (7% annual interest rate), and \(t\) is time in years.

Step 2: Substitute Values

Substitute the given values into the equations to get the models describing the growth of each investment:- Annually: \(A = 500(1 + 0.07/1)^{1*t}\)- Quarterly: \(A = 500(1 + 0.07/4)^{4*t}\)- Continuously: \(A = 500 e^{0.07*t}\)

Step 3: Graph the Models

Use a graphing utility to plot these three models over a 20-year period. Label each graph accordingly.

Step 4: Determine Highest Return

Inspect the graphs at \(t=20\) to see which investment gives the highest return. This is where the y-coordinate, representing the amount of money, is highest.

Step 5: Calculating Differences in Earnings

Calculate the differences in earnings among the three investments by subtracting the lesser amounts from the highest one. This will tell you the differences in earnings after 20 years.

Key Concepts

annually compounded interest

Annually compounded interest means that the interest is calculated and added to the principal once a year. This is the simplest form of compounding and a great starting point for understanding compound interest. The formula for calculating annually compounded interest is: \[ A = P (1 + r)^t \] where:

• \(A\) is the amount of money accumulated after n years, including interest.
• \(P\) is the principal amount (initial investment), in this exercise, it's \($500\).
• \(r\) is the annual interest rate (decimal), in this case, \(0.07\).
• \(t\) is the time in years.

For example, in the exercise, if you want to know the amount after 20 years, you would substitute \(P = 500\), \(r = 0.07\), and \(t = 20\) into the formula. This gives you the equation:\[ A = 500(1 + 0.07)^{20} \]This will show how savings grow over time with interest added once each year.

quarterly compounded interest

Quarterly compounded interest involves calculating and adding the interest to the principal four times a year, or every quarter. This method allows your investment to grow faster than annually due to more frequent compounding periods. The formula for quarterly compounding is:\[ A = P (1 + \frac{r}{4})^{4t} \]Explanation:

• The interest is divided by 4 to account for the four quarterly periods.
• The exponent \(4t\) reflects the total number of compounding periods over t years.

If you were to substitute the same values from the given exercise, it would look like:\[ A = 500(1 + \frac{0.07}{4})^{4 \times 20} \]Notice that with this formula, because interest is compounded more frequently during the year, the money grows a little faster compared to the annually compounded method. By plotting this equation, you can visualize how this impacts the total accrued amount over time.

continuously compounded interest

Continuously compounded interest is a special case where interest is added continuously rather than at discrete intervals like annually or quarterly. This method uses the mathematical constant \(e\) to model exponential growth. The formula for continuously compounded interest is:\[ A = P e^{rt} \]In this formula:

• The constant \(e\) is approximately equal to 2.71828.
• \(rt\) represents the rate multiplied by time.

For this exercise, you would substitute the values as follows:\[ A = 500 e^{0.07 \times 20} \]This approach results in the highest amount of growth potential when compared to annually or quarterly compounded interest due to the nature of continuous compounding. In practice, while the difference might seem small over shorter periods, it becomes more pronounced over longer periods like the 20 years described here. This is why, when plotted, this formula tends to yield a higher amount than the other methods.