Assume that the number of bacteria follows an exponential growth model:
P(t)=P0e^k/t. The count in the bacteria culture was 400 after 10 minutes and 1500 after 35 minutes.
(a) What was the initial size of the culture?
(b) Find the population after 85 minutes.
(c) How many minutes after the start of the experiment will the population reach 12000?
P = Po e^kt
400 = Po e^10k
1500 = Po e^35 t
1/Po = (1/400) e^10 k = (1/1500)e^35 k
e^35 k / e^10k = 1500/400 = 15/4
e^25 k = 15/4
25 k = ln(15/4) = 1.322
k = .0529
so
P = Po e^.0529 t
400 = Po e^.529
Po = 236
then
P = 236 e^.0529 t
I think you can take it from there
I think you have a typo or a flaw in your equation.
P(t) = P0 e^(k/t) makes no sense at t = 0
and since you are asking for the initial value P0, that would assume we need t = 0 in our calculation.
That would make our result undefined.
Fix your problem.
Did not see Damon's answer, I had the page open all this time.
To determine the initial size of the culture (P0), we can use the given information that the count was 400 after 10 minutes. We are given the equation for exponential growth: P(t) = P0e^(k/t), where P(t) is the population at time t, P0 is the initial population, k is the growth rate constant, and t is the time.
(a) Given that P(10) = 400, we substitute these values into the equation:
400 = P0e^(k/10).
Next, we can determine the growth rate constant, k, by using the information that the count was 1500 after 35 minutes.
Using P(35) = 1500 in the equation, we get:
1500 = P0e^(k/35).
Now we have a system of equations:
400 = P0e^(k/10)
1500 = P0e^(k/35).
To solve this system, we can divide the two equations:
(400/1500) = (P0e^(k/10))/(P0e^(k/35)).
Simplifying, we have:
0.267 = e^((k/10) - (k/35)).
To solve for k, we can take the natural logarithm (ln) of both sides:
ln(0.267) = (k/10) - (k/35).
Solving for k gives us:
k/10 - k/35 = ln(0.267).
Combining similar terms, we get:
(35k - 10k)/(10 * 35) = ln(0.267).
(25k)/(350) = ln(0.267).
Finally, solving for k, we have:
k = (350 * ln(0.267))/25.
Now we can substitute the value of k back into one of the original equations to find P0.
Using P(10) = 400, we get:
400 = P0e^((350 * ln(0.267))/(25 * 10)).
Simplifying, we have:
400 = P0e^(350 * ln(0.267))/250.
4/1 = P0 * 0.43069.
Finally, solving for P0, we get:
P0 = 4/0.43069 = 9.282.
Therefore, the initial size of the culture (P0) is approximately 9.282.
(b) To find the population after 85 minutes, we substitute the values into the equation:
P(85) = P0e^(k/85).
Using the calculated values of P0 and k, we have:
P(85) = 9.282 * e^((350 * ln(0.267))/25 * 85).
Calculating this expression will give us the population after 85 minutes.
(c) To determine how many minutes it will take for the population to reach 12,000, we need to solve for t in the equation:
12,000 = P0e^(k/t).
Using the calculated P0 and k values, we rearrange the equation to solve for t:
12,000/9.282 = e^((350 * ln(0.267))/25) / e^(k/t).
Simplifying, we have:
1293.75 = e^((350 * ln(0.267))/25) / e^((350 * ln(0.267))/t).
To isolate t, we can take the natural logarithm (ln) of both sides:
ln(1293.75) = ((350 * ln(0.267))/25) - ((350 * ln(0.267))/t).
Next, we can solve for t:
(350 * ln(0.267))/t = ((350 * ln(0.267))/25) - ln(1293.75).
Finally, solving for t, we get:
t = (350 * ln(0.267))/(((350 * ln(0.267))/25) - ln(1293.75)).
Calculating this expression will determine the number of minutes after the start of the experiment when the population reaches 12,000.