Question 354040
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The future value, *[tex \Large A], of a present value, *[tex \Large P], invested at *[tex \Large 100r] percent interest for *[tex \Large t] years compounded continuously is given by:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ A\ =\ Pe^{rt}]


For the investment to triple, *[tex \Large \frac{A}{P}\ =\ 3], but since *[tex \Large \frac{A}{P}\ =\ e^{rt}], since we know that *[tex \Large r\ =\ \frac{16}{100}], we just need to solve:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ e^{0.16t}\ =\ 3]


for *[tex \Large t]


Take the natural log of both sides:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \ln(e^{0.16t})\ =\ \ln(3)]


Use *[tex \LARGE \log_b{x^n}\ =\ n\log_b(x)] and *[tex \LARGE \log_b(b)\ =\ 1]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 0.16t\ln(e)\ =\ \ln(3)]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ t\ =\ \frac{ln(3)}{.016}]


The rest is just calculator work.


John
*[tex \LARGE e^{i\pi} + 1 = 0]
My calculator said it, I believe it, that settles it
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