Question

In 2004, a specific species of birds had a population of 1000. About three years later, the population was 216. Write an exponential function p(t)=a(b)^t that represents the population t years after 2004.

Answer 1

`\boxed{p(t) = 1000 (0.6)^t}`

Explanation:

We can model the population decline of the birds as

`p(t) = a b^t`

where
a is the initial population
t = number of years elapsed
p(t) is the population after t years

We are given a = 1000 which is the initial population at year 2004

After t = 3 years, the population would be

p(3) and we are given this figure as 216

Plugging in knowns into
`p(t) = a b^t`

`216 = 1000\cdot b^3`

Switch sides so variable is on left

`1000\cdot b^3 = 216\\\\b^3 = (216)/(1000)\\\\\\b = \sqrt[3]{(216)/(1000)} }\\\\\\\\b = \frac{\sqrt[3]{216} }{\sqrt[3]{1000} }\\\\\\b = (6)/(10)\\\\b = 0.6`

Therefore the exponential function that represents the population t years after 2004 is

`\boxed{p(t) = 1000 (0.6)^t}`