# The number of bacteria in a culture grew from 275 to 1135 in three hours. How do you find the number of bacteria after 7 hours?

May 19, 2017

$7381$

#### Explanation:

Bacteria undergo asexual reproduction at an exponential rate. We model this behavior using the exponential growth function.

color(white)(aaaaaaaaaaaaaaaaaa)color(blue)(y(t) = A_(o)*e^(kt)

Where

• $\text{y("t") = value at time ("t")}$
• A_("o") = "original value"
• $\text{e = Euler's number 2.718}$
• $\text{k = rate of growth}$
• $\text{t = time elapsed}$

You are told that a culture of bacteria grew from color(red)[275 to color(red)[1135 in $\textcolor{red}{\text{3 hours}}$. This should automatically tell you that:

• color(blue)[A_("o") = $\textcolor{red}{275}$

• color(blue)["y"("t")] = $\textcolor{red}{\text{1135}}$, and

• $\textcolor{b l u e}{\text{t}}$ = $\textcolor{red}{\text{3 hours}}$

Let's plug all this into our function.

$\textcolor{w h i t e}{a a a a a a a a a a} \textcolor{b l u e}{y \left(t\right) = {A}_{o} \cdot {e}^{k t}} \to \textcolor{red}{1135} = \left(\textcolor{red}{275}\right) \cdot {e}^{k \cdot \textcolor{red}{3}}$

We can work with what we have above because we know every value except for the $\text{rate of growth", color(blue)[k]}$, for which we will solve.

$\textcolor{w h i t e}{- -}$

$\underline{\text{Solving for k}}$

• $\textcolor{red}{1135} = \left(\textcolor{red}{275}\right) \cdot {e}^{k \cdot \textcolor{red}{3}}$

• $\stackrel{\text{4.13}}{\cancel{\frac{\left(1135\right)}{\left(275\right)}}} = \cancel{\frac{275}{275}} {e}^{k \cdot 3}$

• $4.13 = {e}^{k \cdot 3}$

• ${\textcolor{w h i t e}{a}}_{\ln} 4.13 = {\textcolor{w h i t e}{a}}_{\cancel{\ln}} \left({\cancel{e}}^{k \cdot 3}\right)$

• $1.42 = k \cdot 3$

• stackrel"0.47"cancel[((1.42))/((3))] = k*cancel[(3)/(3)

• $0.47 = k$

Why did we figure all this out? Didn't the question ask to solve for the number of bacteria after $\text{time = 7 hours}$ and not for $\textcolor{b l u e}{k} , \text{the rate of growth}$?

The simple answer is that we needed to figure out the $\text{rate of growth}$ so that from there we can figure out the value at time $\left(7\right)$ by setting up a new function since we will have only 1 unknown left to solve.
$\textcolor{w h i t e}{- -}$

$\underline{\text{Solving for number of bacteria after 7 hours}}$

$\textcolor{b l u e}{y \left(t\right) = {A}_{o} \cdot {e}^{k t}} \to y = \left(275\right) \cdot {e}^{0.47 \cdot 7}$

$y = \left(275\right) \cdot {e}^{3.29}$

$y = \left(275\right) \cdot \left(26.84\right)$

$y = 7381$

So, the bacteria colony will grow to $7381$ in number after $\text{7 hours}$