A bacterial population in a broth culcture has been growing exponentially. It was at a density of 4300 individuals per mL two hours ago and is now 5850 individuals per mL. What was the density 4 hours and 15 minutes ago?
Let me amplify:
5850 = 4300 e^(rate*time) and time is 2 hrs
1/2 ln(5850/4300)=rate
rate=.154
Now, 4 .25 hrs ago
5850 =N e^.154*4.25 taking ln of each side
ln5850=ln(N) + .6545
ln N=8.674-6545
N=e^8.0105
N=3012
5850 = 4300 e^(8r) ... ln(5850/4300) = 8 r
5850 = d e^(17r)
ascac
To find the bacterial population density 4 hours and 15 minutes ago, we will need to use the exponential growth formula. The formula for exponential growth is:
N = N0 * e^(kt),
where:
N = final population density,
N0 = initial population density,
e = base of natural logarithm (approximately 2.718),
k = exponential growth rate,
t = time elapsed.
In this case, we have the final population density N = 5850 individuals per mL and the initial population density N0 = 4300 individuals per mL. We need to find the value of t at which the population density was N0.
Let's follow the following steps to find the exponential growth rate k:
1. Calculate the value of t:
Since the bacteria were growing for 2 hours, the value of t is 2 hours. We need to convert the additional 15 minutes to hours. There are 60 minutes in an hour, so 15 minutes is equivalent to 15/60 = 0.25 hours. Therefore, the total time elapsed t is 2 hours + 0.25 hours = 2.25 hours.
2. Plug the values into the exponential growth formula:
5850 = 4300 * e^(k * 2.25).
3. Divide both sides of the equation by 4300:
5850 / 4300 = e^(k * 2.25).
4. Take the natural logarithm (ln) of both sides:
ln(5850 / 4300) = k * 2.25.
5. Solve for k:
k = ln(5850 / 4300) / 2.25.
Now that we have the value of k, we can find the population density 4 hours and 15 minutes ago by plugging this value of k into the exponential growth formula and using the value of t = 4 hours + 0.25 hours = 4.25 hours:
N = N0 * e^(k * t).
Let's calculate this value:
N = 4300 * e^(k * 4.25).
The calculated result will give us the population density 4 hours and 15 minutes ago.