Estimate population as a function?

Question

The population of a colony of rabbits grows exponentially. The colony begins with 5 rabbits; 5 years later there are 370 rabbits.

 

(a): Express the population of the colony of rabbits, P, as a function of time, t, in years. 

 

P(t)= ???

 

(b): Use the graph to estimate how long it takes for the population of rabbits to reach 1000 rabbits.
It will take ????? years. (round to nearest 0.01 year)

Answer 1

f(x) = ab^x, where a = initial amount, b = ratio of growth, x = time

 

370 = 5b⁵

74 = b⁵

⁵√74 = b

74^1/5 = b

 

P(t) = 5(74^1/5)^t

 

1000 = 5(74^1/5)^t

1000 = 5(74^t/5)

200 = 74^t/5

ln 200 = ln 74^t/5

ln 200 = (t/5) ln 74

5.298 ≈ (t/5) (4.30)

26.46 ≈ 4.30 t

6.15 ≈ t

Answer 2

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