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Chapter 4: Problem 24

Find a formula for estimating how long money takes to increase by a factor of ten at \(R\) percent annual interest compounded continuously.

Short Answer

Expert verified
The formula to estimate the time it takes for money to increase by a factor of ten at R% annual interest compounded continuously is: \(t = \frac{ln(10)}{R \div 100}\)

Step by step solution

01

Understand the formula for continuous compounding

The formula for continuous compounding is given by the equation: \(A(t) = P * e^{rt}\) Where: - \(A(t)\) is the final amount at time t, - P is the initial principal, - r is the annual interest rate (expressed as a decimal), - t is the time in years, and - e is the base of the natural logarithm (approximately 2.718).
02

Rewrite the formula for a tenfold increase

Since we want to find the time it takes for the initial amount to increase by a factor of ten, it implies that: \(10 * P = P * e^{rt}\) We can observe that the initial principal (P) appears on both sides of the equation. This allows us to simplify our equation.
03

Simplify the equation

Dividing both sides of the equation by P gives us: \(10 = e^{rt}\) Now we want to solve for t, but first, we need to get rid of the exponential function (e). For which, we will take the natural logarithm (ln) of both sides of the equation.
04

Take the natural logarithm of both sides

Applying the natural logarithm on both sides: \(ln(10) = ln(e^{rt})\) By using the property of logarithms that ln(a^b) = b*ln(a), the equation becomes: \(ln(10) = rt\) Now, we just need to isolate the time variable (t) by dividing both sides by the given interest rate (r).
05

Solve for the time variable (t)

To find the time it takes for the initial amount to increase ten times, we get: \(t = \frac{ln(10)}{r}\) Finally, we have found the formula to calculate the time it takes for the initial amount to increase by a factor of ten at a given interest rate (R%). By converting the annual interest rate from a percentage to decimal (R% 梅 100), we can plug the value (r) into our formula to estimate the time (t). Formula to estimate the time it takes for money to increase by a factor of ten at R% annual interest compounded continuously: \(t = \frac{ln(10)}{R \div 100}\)

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