# Find the number of bacteria after dividing

Pink eye is an infection casued by bacteria. After you have been infected, these bacteria double every 40 minutes. How many bacteria would be present at 6 pm if a single bacterium began reproducing at 11am?

$a=1$ and $r=2$ for this geometric sequence and every $40$ min = $2 \over 3$ hrs, so I've come up with the equation

\begin{align} t_n&=a(r)^{n-1} \\ t_n&=(2)^{{3\over2}(n-1)} \end{align}

$7$ hours passed, so $n=8$

\begin{align} t_n&=(2)^{{3\over 2}(8-1)} \\ &=2^{{3\over2}(7)} \\ &=2^{21\over2} \\ &=1448.1546 \end{align}

So, there will be $1448$ bacteria.

However, this is wrong. The answer is supposed to be $724$.

Why is my answer wrong? I double checked by creating a chart, and $724$ does not seem reasonable:

$$\begin{array}{|c|c|} \hline \text{Time} & \text{Number of Bacteria} \\\hline 11:00 \text{ am} & 1\\\hline 11:40 \text{ am} & 2\\\hline 12:20 \text{ pm} & 4\\\hline 1:00 \text{ pm} & 8\\\hline 1:40 \text{ pm} & 16\\\hline 2:20 \text{ pm} & 32\\\hline 3:00 \text{ pm} & 64\\\hline 3:40 \text{ pm} & 128\\\hline 4:20 \text{ pm} & 256\\\hline 5:00 \text{ pm} & 512\\\hline 5:40 \text{ pm} & 1024\\\hline 6:00 \text{ pm} & \text{between }1024 \text{ and } 2048 \\\hline 6:20 \text{ pm} & 2048 \\\hline \end{array}$$

Even at 5:40 pm, there should be more than $724$ bacteria. Was there something wrong in my solution or is the correct answer wrong?

• Your answer is right. I presume the questioner made a 'fence-post' error. – Penguino May 8 '18 at 21:42
• Yes, a mistake 724 = 2^(19/2) – Phil H May 8 '18 at 22:15
• @Penguino Could you please post your comment as an answer so this question can be removed from the "Unanswered" queue? – Robert Howard Jul 29 '18 at 22:24
• I will add my concurrence that your table is right and the book wrong - in order to say that the question is ridiculous biology. I hate fake "real world problems" that suggest that they are applications of mathematics. – Ethan Bolker Jul 29 '18 at 23:50

Your table is correct until $6:00$ PM. As I read the question the doubling is instantaneous, so at $6:00$ PM there are still $1024$ bacteria. Neither you nor the solution manual should add up all the entries because your table shows the number of bacteria at each time after the doubling. The answer should be $1024$ bacteria at $6:00$PM.