SOLUTION: The population of a colony of rabbits grows exponentially. The colony begins with 5 rabbits; 5 years later there are 320 rabbits. (a) estimate how long it takes for the population

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: The population of a colony of rabbits grows exponentially. The colony begins with 5 rabbits; 5 years later there are 320 rabbits. (a) estimate how long it takes for the population      Log On


   

Question 1206832: The population of a colony of rabbits grows exponentially. The colony begins with 5 rabbits; 5 years later there are 320 rabbits.
(a) estimate how long it takes for the population of rabbits to reach 1000 rabbits.
Found 3 solutions by josgarithmetic, MathLover1, Edwin McCravy:
Answer by josgarithmetic(39618) About Me  (Show Source):
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Model as y=ab%5Ex

ab%5Ex=y
5b%5E5=320
b%5E5=64
b=64%5E%281%2F5%29
b=64%5E%280.2%29
b=2.2974

Model in this form is highlight_green%28y=5%282.2974%29%5Ex%29.
.
.
2.2974%5Ex=1000%2F5
log%28%282.2974%5Ex%29%29=log%28%28200%29%29
x=log%28%282.2974%29%29=log%28%282%2A100%29%29
x=log%28%282%2A100%29%29%2Flog%28%282.2974%29%29
x=%28log%28%282%29%29%2Blog%28%28100%29%29%29%2Flog%28%282.2974%29%29
x=%28log%28%282%29%29%2B2%29%2Flog%28%282.2974%29%29
x=6.369869
or x, about 6 years 4 months

Answer by MathLover1(20850) About Me  (Show Source):
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let population og rabbits be p, time t (in years)

p%28t%29=a%2Ae%5E%28b%2At%29

if the colony begins with 5+rabbits, p%280%29=5
5=a%2Ae%5E%28b%2A0%29
5=a%2A1
a=5

so far equation is
p%28t%29=5%2Ae%5E%28b%2At%29

if 5 years later there are 320 rabbits
320=5%2Ae%5E%28b%2A5%29
320%2F5=e%5E%28b%2A5%29
e%5E%28b%2A5%29=64...take natural log of both sides
ln%28e%5E%28b%2A5%29%29=ln%2864%29
%28b%2A5%29ln%28e%29=ln%2864%29
%28b%2A5%29%2A1=4.1588830833596715
b=4.1588830833596715%2F5
b=0.8317766166719343

equation is:
p%28t%29=5%2Ae%5E%280.8317766166719343%2At%29

(a) estimate how long it takes for the population of rabbits to reach 1000 rabbits.
1000=5%2Ae%5E%280.8317766166719343%2At%29
e%5E%280.8317766166719343%2At%29=200
ln%28e%5E%280.8317766166719343%2At%29%29=ln%28200%29
%280.8317766166719343%2At%29%2Aln%28e%29=ln%28200%29
0.8317766166719343%2At=5.298317366548
t=5.298317366548%2F0.8317766166719343
t=6.37+years

Answer by Edwin McCravy(20056) About Me  (Show Source):
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P%5Bn%5D%22%22=%22%22P%5B0%5De%5E%28rt%29

P%5Bn%5D%22%22=%22%225e%5E%28rt%29

P%5B5%5D%22%22=%22%225e%5E%28r%2A5%29

320%22%22=%22%225e%5E%285r%29

64%22%22=%22%22e%5E%285r%29

ln%2864%29%22%22=%22%225r

ln%2864%29%2F5%22%22=%22%22r

P%5Bn%5D%22%22=%22%225e%5E%28rt%29

1000%22%22=%22%225e%5E%28rt%29

200%22%22=%22%22e%5E%28rt%29

ln%28200%29%22%22=%22%22rt

ln%28200%29%2Fr%22%22=%22%22t

ln%28200%29%5E%22%22%2F%28ln%2864%29%2F5%29%22%22=%22%22t

5ln%28200%29%2F%28ln%2864%29%29%22%22=%22%22t

6.369880158%22%22=%22%22t

About 6.4 years or 6 years, 5 months.

Edwin