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Answer :
To find the population of bacteria in the culture after 13 hours, we can use the formula for exponential growth:
[tex]\[ P_t = P_0 \cdot 2^{\frac{t}{d}} \][/tex]
Where:
- [tex]\( P_0 \)[/tex] is the initial population of bacteria.
- [tex]\( t \)[/tex] is the time in hours.
- [tex]\( d \)[/tex] is the doubling time.
- [tex]\( P_t \)[/tex] is the population after [tex]\( t \)[/tex] hours.
In this problem:
- The initial population [tex]\( P_0 \)[/tex] is 230 bacteria.
- The time [tex]\( t \)[/tex] is 13 hours.
- The doubling time [tex]\( d \)[/tex] is 9 hours.
Plug these values into the formula:
[tex]\[ P_t = 230 \cdot 2^{\frac{13}{9}} \][/tex]
1. First, calculate [tex]\( \frac{13}{9} \)[/tex], which gives approximately 1.444.
2. Next, calculate [tex]\( 2^{1.444} \)[/tex]. This step will give you the growth factor over 13 hours.
3. Multiply this result by the initial population of 230.
Once these calculations are made, you find that the population of bacteria after 13 hours is approximately 626, 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 of bacteria.
- [tex]\( t \)[/tex] is the time in hours.
- [tex]\( d \)[/tex] is the doubling time.
- [tex]\( P_t \)[/tex] is the population after [tex]\( t \)[/tex] hours.
In this problem:
- The initial population [tex]\( P_0 \)[/tex] is 230 bacteria.
- The time [tex]\( t \)[/tex] is 13 hours.
- The doubling time [tex]\( d \)[/tex] is 9 hours.
Plug these values into the formula:
[tex]\[ P_t = 230 \cdot 2^{\frac{13}{9}} \][/tex]
1. First, calculate [tex]\( \frac{13}{9} \)[/tex], which gives approximately 1.444.
2. Next, calculate [tex]\( 2^{1.444} \)[/tex]. This step will give you the growth factor over 13 hours.
3. Multiply this result by the initial population of 230.
Once these calculations are made, you find that the population of bacteria after 13 hours is approximately 626, when rounded to the nearest whole number.
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