Answer:
Part 1) [tex]k=1,000,r=1.2,p=1,000(1.2^t)[/tex]
Part 2) The population after 10 years is about 6,192 humans
Step-by-step explanation:
Part 1) Find the values of k and r
Let
p ----> the human population of a remote island
t ----> the time in years
we have
[tex]p=k(r^t)[/tex]
This is a exponential function of the form [tex]y=a(b^x)[/tex]
where
a is the initial value or y-intercept
b is the base of the exponential function
In this problem
The initial population is 1,000
so
[tex]k=1,000[/tex]
[tex]p=1,000(r^t)[/tex]
Remember that
the population is 1200 after one year
we have the ordered pair (1,1,200)
substitute in the equation and solve for r
[tex]1,200=1,000(r^1)\\1,000r=1,200\\r=1.2[/tex]
therefore
[tex]p=1,000(1.2^t)[/tex]
Part 2) What is the population after 10 years?
For x=10 years
substitute the value of x in the exponential function
[tex]p=1,000(1.2^{10})\\p=6,192[/tex]
therefore
The population after 10 years is about 6,192 humans
