1. ## [SOLVED] exponential function

I've been trying to do this exercise for a while , and the only result I've gotten is 440. Can someone help me. Thanks in advance!

At the beginning of an experiment, a culture contains 200 bacteria. An hour later there are 205 bacteria. Assuming that the bacteria grow exponentially, How many will there be after 2 day?

2. Originally Posted by vance
I've been trying to do this exercise for a while , and the only result I've gotten is 440. Can someone help me. Thanks in advance!

At the beginning of an experiment, a culture contains 200 bacteria. An hour later there are 205 bacteria. Assuming that the bacteria grow exponentially, How many will there be after 2 day?
The quantity of bacteria at time $\displaystyle t$ should be of the form $\displaystyle f(t)=ke^{at}.$ Let's use hours as the unit for $\displaystyle t\text.$

From the problem statement, we have

$\displaystyle f(0) = 200\Rightarrow ke^{0a}=200\Rightarrow k=200\text.$

So $\displaystyle f(t) = 200e^{at}\text.$ Then, from the second condition given, we know that

$\displaystyle f(1)=205\Rightarrow200e^a=205\Rightarrow e^a=\frac{205}{200}=\frac{41}{40}$

$\displaystyle \Rightarrow a=\ln\left(\frac{41}{40}\right)\text.$

Thus, we have $\displaystyle f(t)=200e^{t\ln(41/40)}=200\left(\frac{41}{40}\right)^t,$ so

$\displaystyle f(48) = 200\left(\frac{41}{40}\right)^{48}\approx654.30\te xt{ bacteria.}$

3. Originally Posted by vance
I've been trying to do this exercise for a while , and the only result I've gotten is 440. Can someone help me. Thanks in advance!

At the beginning of an experiment, a culture contains 200 bacteria. An hour later there are 205 bacteria. Assuming that the bacteria grow exponentially, How many will there be after 2 day?
First, remember that your times must be in the same units. In other words, either use days or hours. You are given times of 1 hour and 2 days. You can either calculate how many days 1 hour is equal to or how many hours are in two days. I would personally use the latter.

$\displaystyle f(t) = Pe^{rt}$

$\displaystyle f(0) = 200e^{0r} = 200$

You need to figure out the rate.

$\displaystyle 205 = 200e^{1r}$

$\displaystyle 1.025 = e^{r}$

$\displaystyle r = ln(\frac{205}{200})$

So for 48 hours:

$\displaystyle f(48) = 200e^{(\frac{205}{200})(48)}$

$\displaystyle f(48) = 654.3$

4. Thanks Guys!!!

5. You don't HAVE to use "e" for exponential problems. Saying that the growth is exponential tells us that $\displaystyle f= ab^t$. Since $\displaystyle b^0= 1$ for all b, $\displaystyle f(0)= a= 200$. Taking t in hours, when t= 1 we have [tex]f(t)= 200b= 205[tex] so $\displaystyle b= \frac{205}{200}= \frac{41}{40}$. After 48 hours, there are
$\displaystyle f(48)= 200b^{48}= 200b^{48}= 200(\frac{41}{40})^{48}= 200(3.27148956)= 654$ bacteria.