Suppose that at time t (hours), the number of bacteria in a culture is given by
Re) = 4000 e0.24. How many bacteria are in the culture after six hours? How
long will it take for the bacteria count to reach 100.0007
I think you mean
R(t) = 4000 e^(0.24t)
If so, then
R(6) = 4000 e^1.44 = 16883
Then you want t such that
4000 e^(0.24t) = 100,000 (I assume that the 7 is a typo)
e^(0.24t) = 25
0.24t = ln25
t = ln25/0.24 = 13.41
To find the number of bacteria in the culture after six hours, you can substitute the given time value (t = 6) into the equation. Here's how you can do it step by step:
Step 1: Start with the given equation: Re) = 4000e^(0.24t).
Step 2: Substitute the time value of 6 into the equation: Re) = 4000e^(0.24*6).
Step 3: Simplify the calculation inside the parentheses: Re) = 4000e^(1.44).
Step 4: Evaluate the exponential term using a calculator: Re) ≈ 4000*(4.228).
Step 5: Perform the final multiplication to get the answer: Re) ≈ 16,912 bacteria.
Therefore, there would be approximately 16,912 bacteria in the culture after six hours.
To calculate how long it will take for the bacteria count to reach 100,000.7, you need to solve the equation for the time (t). Here's how you can proceed:
Step 1: Start with the given equation: Re) = 4000e^(0.24t).
Step 2: Set the equation equal to 100,000.7: 4000e^(0.24t) = 100,000.7.
Step 3: Divide both sides of the equation by 4000 to isolate the exponential term: e^(0.24t) = 100,000.7/4000.
Step 4: Simplify the right side of the equation: e^(0.24t) ≈ 25.017675.
Step 5: Take the natural logarithm (ln) of both sides to eliminate the exponential: ln(e^(0.24t)) = ln(25.017675).
Step 6: Use the property of logarithms (ln(e^x) = x) to simplify the left side of the equation: 0.24t = ln(25.017675).
Step 7: Divide both sides of the equation by 0.24 to solve for t: t ≈ ln(25.017675) / 0.24.
Step 8: Evaluate the right side of the equation using a calculator: t ≈ 6.4348.
Therefore, it would take approximately 6.4348 hours for the bacteria count to reach 100,000.7.
To find the number of bacteria in the culture after six hours, we can substitute t = 6 into the equation Re) = 4000e^0.24.
R(6) = 4000e^(0.24*6)
= 4000e^1.44
Using a calculator, we can calculate e^1.44 ≈ 4.2169650343.
R(6) ≈ 4000 * 4.2169650343
≈ 16,867.86
Therefore, there are approximately 16,867.86 bacteria in the culture after six hours.
To find how long it will take for the bacteria count to reach 100,000.7, we need to solve the equation 100,000.7 = 4000e^0.24t for t.
Divide both sides of the equation by 4000:
100,000.7 / 4000 = e^0.24t
Simplify the left side:
25.017675 = e^0.24t
Take the natural log (ln) of both sides:
ln(25.017675) = ln(e^0.24t)
Using the properties of logarithms, we can bring down the exponent:
ln(25.017675) = 0.24t * ln(e)
Since ln(e) = 1, we can simplify the equation further:
ln(25.017675) = 0.24t
Now, divide both sides by 0.24 to solve for t:
t = ln(25.017675) / 0.24
Using a calculator, we can find:
t ≈ 5.711 hours
Therefore, it will take approximately 5.711 hours for the bacteria count to reach 100,000.7.