Question 230213
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What you need to solve is


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 2C\ =\ C(1.05)^t]


for *[tex \Large t]


Now this is a rather complex calculation involving logarithms.  But a much simpler way to do it is to approximate the answer using the rule of 72.  The approximation will be close enough because you are rounding to the nearest year.


Simply divide 72 by the interest rate in percent.  72 divided by 5 is 14.4.  Rounded to the nearest year is 14.


By the way, a much closer approximation, using logarithms, comes out to 14.21 -- which also rounds to 14.  Here is the log way to do it:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 2C\ =\ C(1.05)^t]


Divide out the original cost, *[tex \Large C].


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 2\ =\ (1.05)^t]


Take the log of both sides (you can use base 10 or natural, it doesn't matter)


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \ln(2)\ =\ \ln\left((1.05)^t\right)]


Use the laws of logs:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \ln(2)\ =\ t\ln(1.05)]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ t\ =\ \frac{ln(2)}{ln(1.05)}]


Now break out the calculator to discover that


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ t\ \approx\ 14.21]



John
*[tex \LARGE e^{i\pi} + 1 = 0]
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