Answer:
Part 1) [tex]p_0=5,937.8\ people[/tex]
Part 2) [tex]23,752\ people[/tex]
Part 3) [tex]1,781\ persons/year[/tex]
Step-by-step explanation:
In this problem we have a exponential growth function of the form
[tex]p(t)=p_0(1+r)^t[/tex]
where
p_0 is the initial population
t is the number of years
r is the rate of change
we have that
The initial population p_0 has doubled in 5 years
so
[tex]p(5)=2p_0[/tex]
substitute in the equation above
[tex]2p_0=p_0(1+r)^5[/tex]
solve for r
[tex]2=(1+r)^5[/tex]
elevated both sides to 1/5
[tex]r=2^{\frac{1}{5}} -1[/tex]
[tex]r=0.1487[/tex]
substitute
[tex]p(t)=p_0(1+0.1487)^t[/tex]
[tex]p(t)=p_0(1.1487)^t[/tex]
Find the value of p_0
Remember that
the population is 9,000 after 3 years
so
substitute
[tex]9,000=p_0(1.1487)^3[/tex]
[tex]p_0=5,937.8\ people[/tex]
so  Â
[tex]p(t)=5,937.8(1.1487)^t[/tex]
What will be the population in 10 years?
For t=10 years
substitute
[tex]p(t)=5,937.8(1.1487)^{10}= 23,752\ people[/tex]
How fast is the population growing at t = 10?
we know that
For t=0 ----> P=5,937.8 people
For t=10 ---> P=23,752 people
so
[tex](23,752-5,937.8)/(10-0)=1,781\ persons/year[/tex]
The question is an illustration of an exponential function
An exponential function is represented as:
[tex]P = P_o * r^t[/tex]
The initial population doubled in 5 years.
So, we have:
[tex]2P_o = P_o * r^5[/tex]
Divide both sides by Po
[tex]2 = r^5[/tex]
Take the 5th roots of both sides
[tex]r = 1.15[/tex]
So, we have:
[tex]P = P_o * 1.15^t[/tex]
The population in 3 years is 9000.
So, we have:
[tex]9000 = P_o * 1.15^3[/tex]
[tex]9000 = P_o * 1.52[/tex]
Divide both sides by 1.52
[tex]P_o =5921[/tex]
So, the initial population is 5921
When t = 10, we have:
[tex]P = P_o * 1.15^t[/tex]
[tex]P = 5921 * 1.15^{10[/tex]
[tex]P = 23954[/tex]
The rate is then calculated as:
[tex]r = \frac{P}{10}[/tex]
[tex]r = \frac{23954}{10}[/tex]
[tex]r = 2395[/tex]
Hence, the population is growing at a rate of 2395 persons per year
Read more about exponential functions at:
https://brainly.com/question/11464095
