# 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

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 $s(t_1) = s_0 e^{\alpha t_1}$.