A bacteria culture contains 200 cells initially and grows at a rate proportional to its size. After half an hour the population has increased to 360 cells.
(a) Find the number of bacteria after t hours.
(b) Find the number of bacteria after 4 hours.
I found part a, P(t)= 200e^(1.2*t)
but when I solve part b it's different than the model answer!
I found the following:
360 = 200 e^(.5k) , where k is a constant
1.8 = e^(.5k)
ln 1.8 = .5k lne
.5k = ln 1.8
k = 1.1755733..
so Number = 200 e^(1.755733t)
when t=4
number = 200 e^4.702293..
= 22039.9
or appr 22040 bacteria
It looks like you rounded of the value of k too early.
Since you are using a calculator anyway, why round off any numbers at all ?
Just use them the way they appear on your device, so there is no need to re-enter any numbers.
I also use the memory locations of my calculators to store any intermediate results.
If we use your value of k = 1.2
Number = 200 e^(4*1.2) = appr 24302 which is significantly different from the answer I got.
Ah, the world of bacteria growth! Let's see what's causing the discrepancy in your calculation.
If the initial population is 200 cells and the growth rate is proportional to the size, then we can use the formula P(t) = P(0) * e^(kt), where P(t) is the population after t hours, P(0) is the initial population, k is the constant of proportionality, and e is Euler's number.
From the given information, we know that after half an hour, the population increases to 360 cells. Plugging these values into the formula, we have:
360 = 200 * e^(0.5k)
To solve for the constant of proportionality k, we can divide both sides by 200 and take the natural logarithm:
ln(360/200) = 0.5k
Simplifying,
ln(1.8) = 0.5k
Now we can solve for k:
k = 2 * ln(1.8)
With the value of k in hand, we can proceed to part b:
P(4) = 200 * e^(4k)
However, there might be a mistake in the model answer given to you. I suspect that the value of k should be 2 * ln(1.8) instead of 1.2. Give it a try with the corrected value, and let's see if we get a different result!
To find the number of bacteria after 4 hours, we can use the equation derived in part (a): P(t) = 200e^(1.2t).
Plugging in t = 4 into the equation, we get:
P(4) = 200e^(1.2 * 4)
= 200e^4.8
Calculating this expression, we find P(4) ≈ 2981.01.
So, the number of bacteria after 4 hours is approximately 2981.
To find the number of bacteria after a given amount of time, we need to use the growth rate proportional to its size.
In the given problem, we are told that after half an hour, the population has increased to 360 cells. This information can be used to determine the growth rate, which we will use to find the number of bacteria after a specified number of hours.
(a) To find the number of bacteria after t hours, we can use the formula:
P(t) = P₀ * e^(kt)
where P(t) is the population at time t, P₀ is the initial population, e is the base of the natural logarithm (approximately 2.71828), k is the growth rate constant, and t is the time in hours.
We are given that the initial population (P₀) is 200 cells, and after half an hour, the population increases to 360 cells. Let's use this information to solve for k:
P(0.5) = 200 * e^(k * 0.5) = 360
Dividing both sides by 200:
e^(k * 0.5) = 360/200 = 1.8
Taking the natural logarithm of both sides:
k * 0.5 = ln(1.8)
k = (ln(1.8))/0.5 ≈ 1.1939
So, the growth rate constant (k) is approximately 1.1939.
Now we can use this growth rate constant to find the number of bacteria after any given time (t).
(b) To find the number of bacteria after 4 hours, we substitute t = 4 into the formula:
P(4) = 200 * e^(1.1939 * 4) ≈ 200 * e^(4.7756) ≈ 10237.3
Therefore, the number of bacteria after 4 hours is approximately 10237.3.
Please double-check your calculations to see if there was an error in your solution for part (b). If you provide more details about your approach or the steps you followed, I can help identify and correct any mistakes.