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Chapter 4: Problem 132

Solve each problem. Ben Franklin's gift of \(\$ 4000\) to the city of Philadelphia in 1790 was not managed as well as his gift to Boston. The Philadelphia fund grew to only \(\$ 2\) million in 200 years. Find the annual rate compounded continuously that would yield this total value.

Short Answer

Expert verified
The annual rate compounded continuously is approximately 3.11%.

Step by step solution

01

Understand the Problem

The gift given is \(4000, it grew to \)2,000,000 in 200 years. We need to find the annual rate compounded continuously that would yield this total value.
02

Identify the Continuous Compounding Formula

The formula for continuous compounding is given by \[ A = P e^{rt} \], where \( A \) is the amount, \( P \) is the principal, \( r \) is the rate, and \( t \) is the time in years.
03

Plug in the Known Values

Substitute \( A = 2000000 \), \( P = 4000 \), and \( t = 200 \) into the formula: \[ 2000000 = 4000 e^{200r} \]
04

Isolate the Exponential Term

Divide both sides by 4000 to isolate the exponential term: \[ 500 = e^{200r} \]
05

Take the Natural Logarithm of Both Sides

To solve for \( r \), take the natural logarithm of both sides: \[ \text{ln}(500) = 200r \]
06

Solve for the Rate \( r \)

Divide both sides by 200 to solve for \( r \): \[ r = \frac{\text{ln}(500)}{200} \]Calculate the numerical value: \[ r = \frac{6.2146}{200} \approx 0.0311 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Growth
Exponential growth describes a process where the amount increases at a constant percentage rate over time. Think about how bacteria might double every few hours in ideal conditions. With money, interest that is compounded continuously means it is added to the principal at an ever-increasing rate. The general formula to model exponential growth is \( A = P e^{rt} \), where \( A \) is the final amount, \( P \) is the initial principal, \( r \) is the rate of growth, and \( t \) is the time period.
Compounding Formulas
Compounding is a method to calculate the amount of interest earned over time. There are different compounding methods, such as annually, semi-annually, quarterly, and continuously. Continuous compounding uses the formula \( A = P e^{rt} \), where the interest is added an infinite number of times per year. This is the most effective because it results in the highest amount.
For example, if you start with \( \$4000 \) and want to find how much it will grow to in 200 years at a particular interest rate, you would use this formula to make the calculation simpler.
Natural Logarithms
Natural logarithms, often written as \( \ln \), are the inverses of the exponential function. They work with a base of 'e', which is approximately equal to 2.71828. When you have an equation in the form \( e^{x} = y \), taking the natural logarithm of both sides will isolate \( x \) on one side. For example, if you have \( 500 = e^{200r} \), you can take the natural log of both sides: \( \ln(500) = 200r \). This step is crucial to solve for rates in continuous compounding problems.
Annual Rate Calculation
To calculate the annual rate compounded continuously, follow these steps:
- Start with the continuous compounding formula: \( A = P e^{rt} \).
- Identify the known values: \( A = 2000000 \), \( P = 4000 \), and \( t = 200 \).
- Plug these values into the formula to get \( 2000000 = 4000 e^{200r} \).
- Solve for the exponential term: \( 500 = e^{200r} \).
- To isolate \( r \), take the natural log of both sides: \( \ln(500) = 200r \).
- Finally, solve for \( r \): \( r = \frac{\ln(500)}{200} \) which gives approximately \( r = 0.0311 \). This result indicates the annual interest rate that would grow \( \$4000 \) to \( \$2,000,000 \) in 200 years with continuous compounding.

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Most popular questions from this chapter

Solve each problem. Because of the Black Death, or plague, the only substantial period in recorded history when the earth's population was not increasing was from 1348 to \(1400 .\) During that period the world population decreased by about 100 million people. Use the exponential model \(P=P_{0} e^{r t}\) and the data from the accompanying table to find the annual growth rate for the period 1400 to 2000 . If the 100 million people had not been lost, then how many people would they have grown to in 600 years using the growth rate that you just found? $$\begin{array}{|c|c|}\hline \text { Year } & \begin{array}{c}\text { World } \\\\\text { Population }\end{array} \\\\\hline 1348 & 0.47 \times 10^{9} \\\1400 & 0.37 \times 10^{9} \\\1900 & 1.60 \times 10^{9} \\\2000 & 6.07 \times 10^{9} \\\\\hline\end{array}$$

Solve each equation. Find the exact solutions. $$\log _{2}\left(\log _{3}\left(\log _{4}(x)\right)\right)=0$$

Power Rule Applying the power rule to \(y=\log \left(x^{2}\right)\) yields \(y=2 \cdot \log (x),\) but are these functions the same? What is the domain of each function? Graph the functions. Find another example of a function whose domain changes after application of a rule for logarithms.

Solve each problem. To illustrate the "miracle" of compound interest, Ben Franklin bequeathed \(\$ 4000\) to the city of Boston in \(1790 .\) The fund grew to \(\$ 4.5\) million in 200 years. Find the annual rate compounded continuously that would cause this "miracle" to happen.

Cooperative Learning Work in a small group to write a summary (including drawings) of the types of graphs that can be obtained for exponential functions of the form \(y=a^{x}\) for \(a>0\) and \(a \neq 1\)
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