Step-by-step explanation:
1. To find the relative rate of growth of the bacteria population, we can use the formula for exponential growth: \( y = ab^t \), where:
- \( y \) is the population at time \( t \),
- \( a \) is the initial population,
- \( b \) is the growth rate, and
- \( t \) is the time elapsed since the initial population.
Given:
- Population after 2 hours, \( y_1 = 400 \),
- Population after 6 hours, \( y_2 = 25600 \),
- Time difference, \( \Delta t = 6 - 2 = 4 \) hours.
We can use these values to find \( b \):
\[ y_2 = ab^{6} \]
\[ 25600 = ab^{6} \]
\[ y_1 = ab^{2} \]
\[ 400 = ab^{2} \]
Now, divide the second equation by the first equation:
\[ \frac{25600}{400} = \frac{ab^{6}}{ab^{2}} \]
\[ 64 = b^{4} \]
Taking the fourth root of both sides:
\[ b = \sqrt[4]{64} \]
\[ b = 2 \]
Now that we have \( b \), we can use it to find \( a \) from either of the original equations. Let's use the first one:
\[ 400 = a(2)^{2} \]
\[ 400 = 4a \]
\[ a = \frac{400}{4} \]
\[ a = 100 \]
Now, we can find the relative rate of growth (\( r \)) as a percentage:
\[ r = (b - 1) \times 100\% \]
\[ r = (2 - 1) \times 100\% \]
\[ r = 100\% \]
So, the relative rate of growth of the bacteria population is 100%.
2. The initial size of the culture (\( a \)) is 100.
3. The function that models the number of bacteria after \( t \) hours is:
\[ y = 100 \times 2^t \]
4. To find the number of bacteria after 4.5 hours, we can plug \( t = 4.5 \) into the function:
\[ y = 100 \times 2^{4.5} \]
Using a calculator, we can find the approximate value of \( y \).
5. To find when the number of bacteria will be 50000, we need to solve for \( t \) in the equation:
\[ 50000 = 100 \times 2^t \]
\[ 500 = 2^t \]
Take the logarithm of both sides:
\[ \log(500) = \log(2^t) \]
\[ \log(500) = t \times \log(2) \]
\[ t = \frac{\log(500)}{\log(2)} \]
Using a calculator, we can find the approximate value of \( t \).
I apologize for the incomplete response. Let's continue with the calculations:
4. To find the number of bacteria after 4.5 hours, we can plug \( t = 4.5 \) into the function:
\[ y = 100 \times 2^{4.5} \]
Using a calculator, we find:
\[ y \approx 100 \times 2^{4.5} \approx 100 \times 22.627 \approx 2262.7 \]
So, the number of bacteria after 4.5 hours is approximately 2262.7.
5. To find when the number of bacteria will be 50000, we need to solve for \( t \) in the equation:
\[ 50000 = 100 \times 2^t \]
\[ 500 = 2^t \]
Take the logarithm of both sides:
\[ \log(500) = \log(2^t) \]
\[ \log(500) = t \times \log(2) \]
\[ t = \frac{\log(500)}{\log(2)} \]
Using a calculator, we find:
\[ t \approx \frac{\log(500)}{\log(2)} \approx \frac{2.699}{0.301} \approx 8.97 \]
So, when the number of bacteria will be 50000, it will be approximately 8.97 hours after the initial count.
