Answer :
To find the population of bacteria in the culture after 13 hours, we use the formula:
[tex]\[ P_t = P_0 \cdot 2^{\frac{t}{d}} \][/tex]
Where:
- [tex]\( P_t \)[/tex] is the population after [tex]\( t \)[/tex] hours.
- [tex]\( P_0 \)[/tex] is the initial population, which is 13,000 bacteria.
- [tex]\( t \)[/tex] is the time in hours, which is 13 hours.
- [tex]\( d \)[/tex] is the doubling time, which is 3 hours.
Here's how you can calculate the population step-by-step:
1. Understand the Formula: The formula [tex]\( P_t = P_0 \cdot 2^{\frac{t}{d}} \)[/tex] calculates the population after [tex]\( t \)[/tex] hours by taking the initial population and multiplying it by 2 raised to the power of [tex]\( \frac{t}{d} \)[/tex]. This accounts for how many doubling periods have occurred in the given time.
2. Substitute the Given Values:
- [tex]\( P_0 = 13000 \)[/tex]
- [tex]\( t = 13 \)[/tex]
- [tex]\( d = 3 \)[/tex]
This gives us the formula:
[tex]\[ P_t = 13000 \cdot 2^{\frac{13}{3}} \][/tex]
3. Calculate the Exponent:
- First, calculate [tex]\( \frac{13}{3} \)[/tex] which is approximately 4.3333.
4. Evaluate the Exponential Expression:
- Calculate [tex]\( 2^{4.3333} \)[/tex], which gives us approximately 20.1595.
5. Calculate the Population:
- Multiply the initial population by this result: [tex]\( 13000 \times 20.1595 \approx 262063.578 \)[/tex].
6. Round to the Nearest Whole Number:
- The population after 13 hours is approximately 262063.578, which rounds to 262,064 when rounded to the nearest whole number.
So, the population of bacteria in the culture after 13 hours is 262,064.
[tex]\[ P_t = P_0 \cdot 2^{\frac{t}{d}} \][/tex]
Where:
- [tex]\( P_t \)[/tex] is the population after [tex]\( t \)[/tex] hours.
- [tex]\( P_0 \)[/tex] is the initial population, which is 13,000 bacteria.
- [tex]\( t \)[/tex] is the time in hours, which is 13 hours.
- [tex]\( d \)[/tex] is the doubling time, which is 3 hours.
Here's how you can calculate the population step-by-step:
1. Understand the Formula: The formula [tex]\( P_t = P_0 \cdot 2^{\frac{t}{d}} \)[/tex] calculates the population after [tex]\( t \)[/tex] hours by taking the initial population and multiplying it by 2 raised to the power of [tex]\( \frac{t}{d} \)[/tex]. This accounts for how many doubling periods have occurred in the given time.
2. Substitute the Given Values:
- [tex]\( P_0 = 13000 \)[/tex]
- [tex]\( t = 13 \)[/tex]
- [tex]\( d = 3 \)[/tex]
This gives us the formula:
[tex]\[ P_t = 13000 \cdot 2^{\frac{13}{3}} \][/tex]
3. Calculate the Exponent:
- First, calculate [tex]\( \frac{13}{3} \)[/tex] which is approximately 4.3333.
4. Evaluate the Exponential Expression:
- Calculate [tex]\( 2^{4.3333} \)[/tex], which gives us approximately 20.1595.
5. Calculate the Population:
- Multiply the initial population by this result: [tex]\( 13000 \times 20.1595 \approx 262063.578 \)[/tex].
6. Round to the Nearest Whole Number:
- The population after 13 hours is approximately 262063.578, which rounds to 262,064 when rounded to the nearest whole number.
So, the population of bacteria in the culture after 13 hours is 262,064.