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Chapter 3: Problem 21

How long does it take for money to triple when compounded continuously at \(5 \%\) per year?

Short Answer

Expert verified
It takes approximately \(21.972\) years for the money to triple when compounded continuously at \(5\%\) per year.

Step by step solution

01

Understand the continuous compound interest formula

The formula for continuously compounded interest is: \(A = P e^{rt}\) Where: - A is the final amount after t years - P is the principal (initial) amount - r is the annual interest rate (in decimal form) - t is the time in years - e is the base of the natural logarithm (approximately 2.71828) In our problem, we want to find the time it takes to triple the initial amount. So, the final amount (A) will be three times the principal (P).
02

Set up the equation

We want to find the time it takes for the money to triple, so we can set up the equation as: \(3P = P e^{0.05t}\)
03

Solve for time (t)

First, divide both sides of the equation by P to isolate the exponential term: \(3 = e^{0.05t}\) Now, we need to solve for t. To do this, take the natural logarithm (ln) of both sides of the equation: \(ln(3) = ln(e^{0.05t})\) Using the property of logarithms, we can simplify the right side of the equation: \(ln(3) = 0.05t \cdot ln(e)\) Since ln(e) is equal to 1, we have: \(ln(3) = 0.05t\) Now, divide both sides by 0.05 to get t: \(t = \frac{ln(3)}{0.05}\)
04

Calculate the value of t

Using a calculator, find the value of t: \(t = \frac{ln(3)}{0.05} 鈮� 21.972\) So, it takes approximately 21.972 years for the money to triple when compounded continuously at 5% per year.

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