To determine the ratio of the number of bacteria at the end of one hour to the number at the beginning, we need to consider how many times the bacteria population doubles within that hour.
Here's a detailed, step-by-step solution:
1. Understand the doubling period:
The population doubles every 12 minutes.
2. Determine how many minutes are in one hour:
There are 60 minutes in one hour.
3. Calculate the number of complete doubling periods within one hour:
Since the bacteria doubles every 12 minutes, we calculate the number of doubling periods in 60 minutes by dividing:
[tex]\[
\frac{60 \text{ minutes}}{12 \text{ minutes/doubling}} = 5 \text{ doublings}
\][/tex]
4. Calculate the population growth factor:
Each time the bacteria population doubles, it multiplies by 2. So, after one doubling period, the population is [tex]\(2\)[/tex] times the initial amount.
- After the first doubling (12 minutes): Population = [tex]\(2 \times\)[/tex] initial population
- After the second doubling (24 minutes): Population = [tex]\(2 \times 2 \times\)[/tex] initial population = [tex]\(2^2 \times\)[/tex] initial population
- After the third doubling (36 minutes): Population = [tex]\(2 \times 2 \times 2 \times\)[/tex] initial population = [tex]\(2^3 \times\)[/tex] initial population
- After the fourth doubling (48 minutes): Population = [tex]\(2^4 \times\)[/tex] initial population
- After the fifth doubling (60 minutes): Population = [tex]\(2^5 \times\)[/tex] initial population
5. Compute the final ratio:
After 5 doublings, the population at the end of one hour is [tex]\(2^5\)[/tex] times the population at the beginning.
[tex]\[
2^5 = 32
\][/tex]
Thus, the ratio of the number of bacteria at the end of the hour to the number of bacteria at the beginning is:
[tex]\[
32:1
\][/tex]
The correct answer is:
[tex]\[
\boxed{32:1}
\][/tex]
