Question

Suppose a single bacterium is placed in a bottle at 11:00 am. It grows and at 11:01 divides into two bacteria. These two bacteria each grow and at 11:02 divide into four bacteria, which grow and at 11:03 divide into eight bacteria, and so on. Now, suppose the bacteria continue to double every minute and the bottle is full at 12:00.How many bacteria are in the bottle at 11:53? What fraction of the bottle is full at that time?Question content area bottomThere will be   enter your response here bacteria in the bottle at 11:53(Type your answer using exponential notation.)part 2 how full will the bottle be?

Answer 1

Given:

A single bacterium is placed in a bottle at​ 11:00 am.

It grows and at​ 11:01 divides into two bacteria.

These two bacteria each grow and at​ 11:02 divide into four​ bacteria.

These four bacteria each grow and at​ 11:03 divide into eight bacteria.

The bottle is full at​ 12:00.

To find:

The number of bacteria are in the bottle at ​11:​53 and the fraction of the bottle is full at that​ time.

Explanation:

After 1 min, the number of bacterias is,

$2^1$

After 2 min, the number of bacterias is,

$2^2$

After 3 min, the number of bacterias is,

$2^3$

In general,

After t min, the number of bacteria is,

$2^t$

If the bottle is full at 12:00 that means after 60 mins, then the number of bacteria is,

$2^{60}$

Therefore, the number of bacteria at 11:53 is,

$2^{53}$

Then the fraction of the bottle is full at 11:53 is,

$\begin{gathered} \frac{Number\text{ of bacteria at 11:53}}{Total\text{ number of bacteria}}=\frac{2^{53}}{2^{60}} \\ =2^{53-60} \\ =2^{-7} \\ =\frac{1}{2^7} \\ =\frac{1}{128} \end{gathered}$

Final answer:

• The number of bacteria at 11:53 is,

$2^{53}$

• The fraction of the bottle is full at 11:53 is,

$\frac{1}{128}$