Question 974849: (Part 1)
The number of trees in a forest is increasing. Every 16 days the number of trees triples. On June 4,there are 236,196 trees.
Write an exponential to represent the total number of trees. Let x represent time in 16 day intervals.
Find the total number of times the tree population has quadrupled when there are 6,377,292 trees.
How long will it be until there are more than 2 billion trees?
Determine how long ago the trees appeared if there was initially 4 trees.
(Part 2)
Now there is a chemical in the air killing trees, approximately 1/4 of the butterflies are dying each week. Local experts have written the equation representing the number of trees at any time as. 18,921(3/4)^x where x is the number of weeks.
How many trees will there be after 7 weeks?
How many weeks will it take for there to be fewer than 100 trees left? Make a table to show how you got your answer.
My algebra teacher doesn't want to help me and I've been trying to work this problem non stop, please help me, this problem is not very easy (for me). Part 1 and 2 are all related. Thank you in advance :)
Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website! y=ab^x is the basic form. When x is increased by 16 days, y triples. 6377292/236196 = almost 27. That is tripling three times
y=a(1+r)^x
a=where it starts
y/a=(1+r)^x
but y/a=3
3=(1+r)^x
Take ln both sides
ln3= x ln (1+r)
1.1=16 ln (1+r)
0.06875=ln (1+r)
raise both to the e power
1.0711=1+r
r=0.0711, the rate of increase.
236296 (1+0.0711)^x ;;is the basic equation.
6377292=236296 (1+0.0711)^x
Divide to get 26.989=1.0711^x
ln of both sides ;; 3.295=x ln 1.0711 ;;; divide by the ln, which is 0.0687
3.295/0.0687 =x ; x=47.96 days or 48 days.
The tree population increases by a factor of 4 in how many days?
ln4=x (ln(1+.0711);;; 1.386=x *0.0687=20.175 days.
It has quadrupled twice and a little more. You can get that de novo by noting it increased by a factor of 27, which is 2 quadruplings but not three.
2,000,000,000=236296 (1.0711)^x
divide out and get 843696=1.0711^x
take logs both sides 9.0435=x (.0687); x= 131.63 days. Note, rounding will make a difference here.
Does this make sense? I get a rough count of 8 triplings, which would be 128 days. Yes, this makes sense.
4=236296(1+.0711)^x
0.00001693=1.0711^x
-10.98= 0.0687 *x
x=160.
Therefore, it was 160 days ago.
That would be 10 triplings. Or 3^10. 4*(3^10)=236296.
============================
18921*(0.75)^x;; x=7
0.75^7= 0.1335
18921*0.1335=2525.65 or 2526 trees
You can do this for x^8, x^9, etc.
or 18921*(0.75)^x=99
99/18721=0.00523
ln(0.00523)=x ln (0.75)
-5.253= x (-0.288)
x= 18.26 weeks.
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