Question 386568
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*[tex \LARGE \ \ \ \ \ \ \ \ \ \ N(t)\ =\ N_o\,\cdot\,2^{\frac{t}{3}}]


We are dealing with *[tex \LARGE N(t)\ =\ 10,000]


Unfortunately, you didn't bother to share the value of *[tex \LARGE N_o]


On the other hand, given the answers you provided, it can only be *[tex \LARGE N_o\ =\ 1].


So, we have


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 2^{\frac{t}{3}}\ =\ \frac{N(t)}{N_o}\ =\ 10,000]


Take the log of both sides (log base is immaterial to the solution -- I use ln)


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \ln\left(2^{\frac{t}{3}}\right)\ =\ \ln(10,000)]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{t}{3}\ln\left(2\right)\ =\ \ln(10,000)]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ t\ =\ \frac{3\,\cdot\,\ln(10,000)}{\ln\left(2\right)}]


The rest is just calculator work.  Round to the nearest integer to get one of your answers.


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