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Answer :
To find the population of bacteria after 14 hours, we will use the given formula:
[tex]\[ P_t = P_0 \cdot 2^{\frac{t}{d}} \][/tex]
where:
- [tex]\( P_0 = 790 \)[/tex] is the initial population.
- [tex]\( t = 14 \)[/tex] is the time in hours.
- [tex]\( d = 5 \)[/tex] is the doubling time.
Here is how the calculation is done step by step:
1. Substitute the values into the formula:
[tex]\[ P_{14} = 790 \cdot 2^{\frac{14}{5}} \][/tex]
2. Calculate the exponent:
[tex]\[ \frac{14}{5} = 2.8 \][/tex]
3. Calculate [tex]\( 2^{2.8} \)[/tex]:
You find the value of [tex]\( 2^{2.8} \)[/tex] using a calculator, which approximately equals 6.964.
4. Multiply the initial population by this value:
[tex]\[ P_{14} = 790 \times 6.964 \][/tex]
5. Complete the multiplication:
[tex]\[ P_{14} \approx 5501.88 \][/tex]
6. Round to the nearest whole number:
[tex]\[ P_{14} \approx 5502 \][/tex]
So, the population of bacteria in the culture after 14 hours is approximately 5502.
[tex]\[ P_t = P_0 \cdot 2^{\frac{t}{d}} \][/tex]
where:
- [tex]\( P_0 = 790 \)[/tex] is the initial population.
- [tex]\( t = 14 \)[/tex] is the time in hours.
- [tex]\( d = 5 \)[/tex] is the doubling time.
Here is how the calculation is done step by step:
1. Substitute the values into the formula:
[tex]\[ P_{14} = 790 \cdot 2^{\frac{14}{5}} \][/tex]
2. Calculate the exponent:
[tex]\[ \frac{14}{5} = 2.8 \][/tex]
3. Calculate [tex]\( 2^{2.8} \)[/tex]:
You find the value of [tex]\( 2^{2.8} \)[/tex] using a calculator, which approximately equals 6.964.
4. Multiply the initial population by this value:
[tex]\[ P_{14} = 790 \times 6.964 \][/tex]
5. Complete the multiplication:
[tex]\[ P_{14} \approx 5501.88 \][/tex]
6. Round to the nearest whole number:
[tex]\[ P_{14} \approx 5502 \][/tex]
So, the population of bacteria in the culture after 14 hours is approximately 5502.
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