Answer :
To determine the population of the bacteria after 13 hours, we can use the population growth formula:
[tex]\[ P_t = P_0 \times 2^{\frac{t}{d}} \][/tex]
where:
- [tex]\( P_t \)[/tex] is the population after [tex]\( t \)[/tex] hours.
- [tex]\( P_0 \)[/tex] is the initial population.
- [tex]\( t \)[/tex] is the time in hours.
- [tex]\( d \)[/tex] is the doubling time.
For this problem:
- The initial population [tex]\( P_0 \)[/tex] is 13,000.
- The doubling time [tex]\( d \)[/tex] is 3 hours.
- The time elapsed [tex]\( t \)[/tex] is 13 hours.
Now, plug these values into the formula:
1. Calculate the exponent: [tex]\(\frac{t}{d} = \frac{13}{3}\)[/tex].
2. Use this result in the expression [tex]\(2^{\frac{13}{3}}\)[/tex].
3. Multiply the initial population by this power of 2:
[tex]\[ P_t = 13,000 \times 2^{\frac{13}{3}} \][/tex]
After you perform the calculations, you will find that:
[tex]\[ P_t \approx 262,064 \][/tex]
Therefore, the population of the bacteria after 13 hours, rounded to the nearest whole number, is 262,064.
[tex]\[ P_t = P_0 \times 2^{\frac{t}{d}} \][/tex]
where:
- [tex]\( P_t \)[/tex] is the population after [tex]\( t \)[/tex] hours.
- [tex]\( P_0 \)[/tex] is the initial population.
- [tex]\( t \)[/tex] is the time in hours.
- [tex]\( d \)[/tex] is the doubling time.
For this problem:
- The initial population [tex]\( P_0 \)[/tex] is 13,000.
- The doubling time [tex]\( d \)[/tex] is 3 hours.
- The time elapsed [tex]\( t \)[/tex] is 13 hours.
Now, plug these values into the formula:
1. Calculate the exponent: [tex]\(\frac{t}{d} = \frac{13}{3}\)[/tex].
2. Use this result in the expression [tex]\(2^{\frac{13}{3}}\)[/tex].
3. Multiply the initial population by this power of 2:
[tex]\[ P_t = 13,000 \times 2^{\frac{13}{3}} \][/tex]
After you perform the calculations, you will find that:
[tex]\[ P_t \approx 262,064 \][/tex]
Therefore, the population of the bacteria after 13 hours, rounded to the nearest whole number, is 262,064.