# Exponential Growth

A bacteria culture is known to grow at a rate proportional to the amount present.After $1$ hour $1000$s stands of the bacteria are observed in the culture;and after $4$years $3000$ strands. Find:

1. an expression for the approximate number of strands of the bacteria present in the culture at any time $t$
2. the approximately number of strands of the bacteria originally in the culture.
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Surely you have an idea about how to begin a solution... –  Did Jul 1 '12 at 22:17
Please try to avoid capslock titles, as they are perceived as though you are shouting. Second, the imperative tone of this post is not fitting the site, please edit it to make it slightly more polite. By the way, if this question is a homework assignment please add the [homework] tag to it. –  Asaf Karagila Jul 1 '12 at 22:17
What have you tried? What equation do you have for exponential growth? –  Ross Millikan Jul 1 '12 at 22:19

Fit the data to the model $s(t) = s_0 e^{\alpha t}$.
Let $t_1 = 1$ hour, $t_2 = 4$ years. You have $s(t_1) = 1000$, $s(t_2) = 3000$.
Then the model gives $s(t_i) = s_0 e^{\alpha t_i}$, $i \in \{1,2\}$.
From this you can easily estimate $\alpha$ using; $\frac{s(t_1)}{s(t_2)} = e^{\alpha (t_2-t_1)}$.
Given $\alpha$, it is straightforward to figure out $s_0$ using $s(t_1) = s_0 e^{\alpha t_1}$.