## anonymous 5 years ago a population of grasshoppers quadruples in 20 days. assuming exponential growth, if the present population is 40 million, what will it be in 50 days. answer by first finding the number of ghoppers as a function of time. y=y0e^kt.

1. anonymous

I devided 50 by 20 and got 2.5 than I multiplied 40 by 4 because I am qudrupaling it 40 times 4+40 times 4+40 times 2: this is because you quadrupal it 2.5 times so it would be: 160+160+80= 400 400 million grasshoppers

2. anonymous

hope this helps:D

3. anonymous

thank you!

4. anonymous

5. anonymous

lets try this: $y=y_0e^{kt}$ $y_0=40$so you have $y=40e^{kt}$

6. anonymous

you do not know k, but you know that when t = 20, y=160 (quadruples) so we write $160=40e^{20k}$ and solve for k

7. anonymous

first step divide by 40, it was unimportant in any case $4=e^{20k}$ $ln(4)=20k$ $k=\frac{ln(4)}{20}=.0693$rounded

8. anonymous

i love how helpful you are thank you so much

9. anonymous

now we go back to the formula and have $y=40e^{.0693t}$ replace t by 50 to get $y=40e^{.0693\times 50}=40e^{3.46574}=40\times 32=1280$\]

10. anonymous

my answer may be off so let me check it

11. anonymous

ok

12. anonymous

i think i made a calculation error somewhere but it is certainly not the answer you were given earlier. doubles every twenty days and when you start counting you have 40 so in 20 days you have 80 and then in another 20 days (40 days later) you have 160

13. anonymous

oh wait. quadruples! ok you start counting you have 40

14. anonymous

yeah i got 160 when i tried on my own, i just got confused

15. anonymous

yeah i got 160 when i tried on my own, i just got confused

16. anonymous

yeah i got 160 when i tried on my own, i just got confused

17. anonymous

in twenty day s you have 160 in another 20 day you have 640

18. anonymous

so where do i plug that in?

19. anonymous

so where do i plug that in?

20. anonymous

so where do i plug that in?

21. anonymous

ok i was right.

22. anonymous

everything i wrote was correct.

23. anonymous

we can do it the easy way without solving like we did, although that is what they want you to do

24. anonymous

first find k, and then replace t by 50

25. anonymous

it is all correct. but lets do it the easy way without all that. you know it quadruples every 20 days. so another formula you can use is $40\times 4^{\frac{t}{20}}$

26. anonymous

now just replace t by 50 and you get $40\times 4^{\frac{50}{20}}=40\times 4^{\frac{5}{2}}$

27. anonymous

$4^{frac{5}{2}}=\sqrt{4^5}=2^5=32$

28. anonymous

and $32\times= 1280$

29. anonymous

i meant 32*40=1280

30. anonymous

either way that is the correct answer

31. anonymous

okay makes sense! thanks again

32. anonymous

welcome