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November 24th, 2015, 10:12 PM  #1 
Newbie Joined: Oct 2015 From: Canada Posts: 5 Thanks: 0  Derivatives, trignometric functions and exponential functions
Hey guys I need help with few questions they are: 1 A dinosaur bone from an archaeological site has onefifth of the amount of Carbon 14 than it originally contained. Determine the approximate age of the bone given that Carbon 14 has a half life of 5770 years. 2 The number of people in a small town became zombies after t days during a zombie apocalypse is given by N(t) and is approximated by the exponential model N(t) =10000/1 + 12.5e^kt If 80 people are zombies by the day 10 , find how many people become zombies by day 15. Thanks 
November 24th, 2015, 11:27 PM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,094 Thanks: 2360 Math Focus: Mainly analysis and algebra 
$c(t) = A\mathrm e^{kt}$ and $c(5770)=\frac12c(0)$. So $$\begin{aligned} A\mathrm e^{5770k} &= \frac12A \\5770k &= \log \frac12 \\ k &= {\log 2 \over 5770} \end{aligned}$$ Then we have $c(0) = \frac15 c(T)$ so$$\begin{aligned} A &= \frac15A\mathrm e^{{\log 2 \over 5770}T} \\ \log 5 &= {\log 2 \over 5770}T \\ T &= {5770\log 5 \over \log 2} \\ &= 13397.5 = 13400 \text{ years} \end{aligned}$$ I think there's something wrong with question 2. 
November 25th, 2015, 07:20 AM  #3  
Math Team Joined: Jan 2015 From: Alabama Posts: 2,879 Thanks: 766  Quote:
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Solve that for k. Start by dividing both sides by 12.5 then take the logarithm of both sides. (Since the number of zombies is increasing, k will be negative.) Quote:
$\displaystyle \frac{10000}{1+ 12.5e^{kt}}$ Quote:
 

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