## mr.luna Group Title 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 2 years ago 2 years ago

1. mr.luna Group Title

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 Group Title

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

3. mr.luna Group Title

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

4. mr.luna Group Title

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

5. mr.luna Group Title

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

6. mr.luna Group Title

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

7. mr.luna Group Title

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

8. victorarana Group Title

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 Group Title

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

10. victorarana Group Title

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

11. victorarana Group Title

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

12. victorarana Group Title

so, $k = \ln4.2$

13. mr.luna Group Title

ok crap .yeah your right! good catch

14. victorarana Group Title

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

15. victorarana Group Title

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

16. victorarana Group Title

sorry its t instead of k

17. mr.luna Group Title

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

18. victorarana Group Title

what did you get?

19. mr.luna Group Title

instead of ln4.2 i get 4.2k

20. victorarana Group Title

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

21. victorarana Group Title

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

22. mr.luna Group Title

oh. ok. got it now. thanks

23. victorarana Group Title

You're welcome.