Problem 21 - Algebra and Trigonometry Real Mathematics, Real People

Question

Complete the table for the time \(t\) (in years) necessary for \(P\) dollars to triple when interest is compounded continuously at rate \(r .\) Create a scatter plot of the data. $$\begin{array}{|c|c|c|c|c|c|c|} \hline r & 2 \% & 4 \% & 6 \% & 8 \% & 10 \% & 12 \% \\ \hline t & & & & & & \\ \hline \end{array}$$

Step-by-Step Solution

Answer

After calculation the table becomes: \n\n\[\begin{array}{|c|c|c|c|c|c|c|}\hline r & 2 \% & 4 \% & 6 \% & 8 \% & 10 \% & 12 \% \ \hline t & 54.60 & 27.30 & 18.20 & 13.65 & 10.92 & 9.10 \ \hline\end{array} \] \n\n The scatter plot will show these values plotted, with the interest rates as the x-values and the times as the y-values.

Step 1: Use the continuously compounded interest formula

First we need to determine the time \(t\) it takes for an amount \(P\) to triple. We can rewrite the continuous compound interest formula as: \(Pe^{rt}=3P\). As we want to find when \(P\) triples, we can simplify this to: \(e^{rt}=3\). Taking the natural logarithm: \(rt = ln(3)\). So the time \(t = ln(3)/r\).

Step 2: Calculate time for each interest rate

Now we need to calculate the time for each interest rate in the table. Using the formula from step 1 and substituting the given interest rate (remembering to convert the percentage to a decimal by dividing by 100) into the formula we can find the correct times for each interest rate. This yields: \(t = ln(3)/0.02, ln(3)/0.04, ln(3)/0.06, ln(3)/0.08, ln(3)/0.10, ln(3)/0.12\).

Step 3: Create Scatter Plot

Use the rates as the x values and the times calculated as the y values to create a scatter plot.

Key Concepts

Compound Interest Formula

The compound interest formula is a mathematical expression used to calculate the amount of interest earned on an investment or loan over some time when interest is compounded at regular intervals. In the case of continuous compounding, the formula transforms into an exponential function represented by \(A = Pe^{rt}\) where:

• \(A\) is the amount of money accumulated after \(n\) years, including interest.
• \(P\) is the principal amount (initial investment).
• \(r\) is the annual interest rate (as a decimal).
• \(t\) is the time in years.
• \(e\) is the base of the natural logarithms and is approximately equal to 2.71828.

Continuous compounding is a powerful concept because, unlike periodic compounding that compounds interest at specific intervals, here, it’s compounded an infinite number of times per period. This results in a faster growth rate of the investment.
In our exercise, we are using this formula to determine the time it takes for an initial investment to triple when the interest is compounded continuously.

Natural Logarithm

The natural logarithm, denoted as \(ln\text{(x)}\), is a type of logarithm with the base \(e\), where \(e\) is the mathematical constant approximately equal to 2.71828. The natural logarithm of x is the power to which \(e\) must be raised to obtain the value x. In other words, if \(y = ln(x)\), then \(e^y = x\). In finance and compounded interest calculations, the natural logarithm is used to isolate the exponent in exponential functions, which is why it plays a central role in solving continuous compound interest problems.
For instance, in the provided exercise, after setting up the equation for continuously compounded interest to find the time it takes for money to triple, \(ln(3)\) is used to obtain the value of \(rt\), which is the product of the rate and the time.

Exponential Functions

Exponential functions are mathematical expressions where the variable appears in the exponent. This type of function describes situations where growth or decay happens at a consistently increasing or decreasing rate, respectively. The general form of an exponential function is \(f(x) = ab^{x}\), where \(a\) is a constant, \(b\) is the base, and \(x\) is the exponent.
In our continuously compounded interest scenario, the principal \(P\) grows exponentially with time \(t\) at a rate \(r\). Exponential growth, represented by the function \(Pe^{rt}\), is common in finance, population growth, and natural processes. Understanding the behavior of exponential functions is crucial for forecasting and simulating real-world situations where change occurs rapidly and, in many cases, increases without an upper bound.

Scatter Plots

Scatter plots are graphical representations used to show the relationship between two different variables, where each pair of values is represented as a point on the plot. In finance, scatter plots can help visualize how changes in one variable, like the interest rate, might affect another variable, such as the time to reach a certain financial goal.
In the explanatory problem at hand, creating a scatter plot with interest rates on the x-axis and the corresponding time \(t\) on the y-axis will allow us to see whether there’s a clear pattern or relationship between the rate of interest and the time taken for the investment to triple. Analyzing such plots can reveal the nature and strength of correlations, trends, and can even lead to predictions about future behavior under similar conditions.