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Chapter 4: Problem 22

How long does it take for money to increase by a factor of five when compounded continuously at \(7 \%\) per year?

Short Answer

Expert verified
It takes approximately \(20.79\) years for the money to increase by a factor of five when compounded continuously at an annual interest rate of \(7\%\).

Step by step solution

01

Understand the continuous compounding formula

The formula for continuous compounding is given by \(A = P \cdot e^{rt}\), where: - \(A\) is the final amount after time \(t\) - \(P\) is the initial principal amount - \(r\) is the annual interest rate - \(t\) is the time in years - \(e\) is the base of the natural logarithm, approximately equal to 2.71828 Since we need to find the time \(t\) for the money to increase by a factor of five, we can write the equation as: \(5P = P \cdot e^{0.07t}\).
02

Solve for time t

Since we need to find the time \(t\), let's start by dividing both sides of the equation by \(P\): \[\frac{5P}{P}= \frac{P \cdot e^{0.07t}}{P}\] \[5 = e^{0.07t}\] Now, we want to get \(t\) out of the exponent. We can do this by applying the natural logarithm, denoted by ln, to both sides of the equation: \[\ln(5)=\ln(e^{(0.07t)})\] Using the logarithmic property, we can bring down the exponent: \[\ln(5) = 0.07t\] Finally, divide both sides by 0.07 to solve for \(t\): \[t=\frac{\ln(5)}{0.07}\]
03

Calculate the time t

Now, compute the value of \(t\): \[t=\frac{\ln(5)}{0.07} \approx 20.79\]
04

Interpret the result

It takes approximately \(20.79\) years for the money to increase by a factor of five when compounded continuously at an annual interest rate of \(7\%\).

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Most popular questions from this chapter

In Section 3.5 we saw that if a radioactive isotope has half-life \(h,\) then the function modeling the number of atoms in a sample of this isotope is $$ a(t)=a_{0} \cdot 2^{-t / h}, $$ where \(a_{0}\) is the number of atoms of the isotope in the sample at time 0 . Many books do not use the formula above but instead use the formula $$ a(t)=a_{0} e^{-(t \ln 2) / h}. $$ Show that the two formulas above are really the same. [Which of the two formulas in this problem do you think is cleaner and easier to understand?]

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