Answer :
To find the population of bacteria in the culture after 13 hours, we can use the formula:
[tex]\[ P_t = P_0 \cdot 2^{\frac{t}{d}} \][/tex]
Where:
- [tex]\( P_0 \)[/tex] is the initial population.
- [tex]\( t \)[/tex] is the time in hours.
- [tex]\( d \)[/tex] is the doubling time in hours.
- [tex]\( P_t \)[/tex] is the population after [tex]\( t \)[/tex] hours.
Given:
- Initial population [tex]\( P_0 = 43,000 \)[/tex].
- Time [tex]\( t = 13 \)[/tex] hours.
- Doubling time [tex]\( d = 5 \)[/tex] hours.
Let's plug these values into the formula to find the population after 13 hours.
1. Substitute the known values into the equation:
[tex]\[ P_t = 43,000 \cdot 2^{\frac{13}{5}} \][/tex]
2. Calculate the exponent [tex]\( \frac{13}{5} \)[/tex]:
[tex]\[ \frac{13}{5} = 2.6 \][/tex]
3. Now, calculate [tex]\( 2^{2.6} \)[/tex].
4. Multiply this result by 43,000 to get the population after 13 hours.
After performing these steps, you will find that the population of bacteria in the culture after 13 hours is approximately 260,703 when rounded to the nearest whole number.
[tex]\[ P_t = P_0 \cdot 2^{\frac{t}{d}} \][/tex]
Where:
- [tex]\( P_0 \)[/tex] is the initial population.
- [tex]\( t \)[/tex] is the time in hours.
- [tex]\( d \)[/tex] is the doubling time in hours.
- [tex]\( P_t \)[/tex] is the population after [tex]\( t \)[/tex] hours.
Given:
- Initial population [tex]\( P_0 = 43,000 \)[/tex].
- Time [tex]\( t = 13 \)[/tex] hours.
- Doubling time [tex]\( d = 5 \)[/tex] hours.
Let's plug these values into the formula to find the population after 13 hours.
1. Substitute the known values into the equation:
[tex]\[ P_t = 43,000 \cdot 2^{\frac{13}{5}} \][/tex]
2. Calculate the exponent [tex]\( \frac{13}{5} \)[/tex]:
[tex]\[ \frac{13}{5} = 2.6 \][/tex]
3. Now, calculate [tex]\( 2^{2.6} \)[/tex].
4. Multiply this result by 43,000 to get the population after 13 hours.
After performing these steps, you will find that the population of bacteria in the culture after 13 hours is approximately 260,703 when rounded to the nearest whole number.