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Chapter 3: Problem 16

Determine the time necessary for \(P\) dollars to double when it is invested at interest rate \(r\) compounded (a) annually, (b) monthly, (c) daily, and (d) continuously. $$r=6.5 \%$$

Short Answer

Expert verified
The time it takes for \(P\) dollars to double when invested at an interest rate of \(6.5\%\), when compounded annually is approximately \(10.63\) years, monthly is approximately \(10.49\) years, daily is approximately \(10.48\) years, and continuously is approximately \(10.67\) years.

Step by step solution

01

Compounded Annually

Use the formula \(A = P(1 + r/n)^{nt}\), set \(A = 2P\) and \(n = 1\), and solve for \(t\). \[t = \frac{\ln(2)}{\ln(1+0.065)} = 10.63 \, years\]
02

Compounded Monthly

Change \(n\) to \(12\) in the annual compound interest formula and solve for \(t\). \[t = \frac{\ln(2)}{12\ln(1+0.065/12)} = 10.49 \, years\]
03

Compounded Daily

With \(n\) set to \(365\), solve for \(t\). \[t = \frac{\ln(2)}{365\ln(1+0.065/365)} = 10.48 \, years\]
04

Compounded Continuously

For continuously compounded interest, set \(A = 2P\) in \(A=Pe^{rt}\) and solve for \(t\). \[t = \frac{\ln(2)}{0.065} = 10.67 \, years\]

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