## mr.luna 4 years ago how do i set this up?! a bacteria culture initially contains 100 dells and grows at a rate proportional to its size. after an hour the population has incresed to 420. a)find an expression for the number of bacteria after t hours. b)find the number of bacteria after 3 hours c)find the rate of growth after 3 hours d)when will the population reach 10,000

1. mr.luna

i have trouble converting it to numbers. here what i have so far and i working on more. i use the P(t)=P(0)e^kt

2. mr.luna

so P(0)=100 P(3)=420

3. mr.luna

i set the equation P(3)=100e^3k=420 i solved and got k=ln42/3

4. mr.luna

is a [100e^[(ln42/3)(t)]

5. mr.luna

b) i just plug in 3 for t to the equation in a) ?

6. mr.luna

for d) i put equation a) equal to 10,000 right?

7. mr.luna

im stuck on c) though. how do you do that?

8. victorarana

I guess that you can derive the expression you got before to obtain dP/dt. And then you sustitute the value of 3 on it. .

9. mr.luna

i got it down to 100(42^t/3) i dont how to dervie?

10. victorarana

I think your initial conditions are wrong. You wrote P(3) = 420, and on the problem says P(1) = 420

11. victorarana

$420 = 100e^{k(1)}$

12. victorarana

so, $k = \ln4.2$

13. mr.luna

ok crap .yeah your right! good catch

14. victorarana

$P = 100e^{\ln(4.2)t}$

15. victorarana

if you derive you get: $\frac{d}{dt}100e^{(\ln 4.2)k}= 100(\ln4.2)e^{(\ln 4.2)k}$

16. victorarana

sorry its t instead of k

17. mr.luna

im trying to get waht you got for the derviative right now. its not coming out for me.

18. victorarana

what did you get?

19. mr.luna

instead of ln4.2 i get 4.2k

20. victorarana

remember that $\frac{d}{dx}e^{ax} = ae^x$

21. victorarana

Sorry i missed an "a" $\frac{d}{dx}e^{ax}=ae^{ax}$

22. mr.luna

oh. ok. got it now. thanks

23. victorarana

You're welcome.