We can model the population of the city using an exponential growth function.
y= a(1+ r)^t
Here a is the initial value and r is the rate of growth. We know that there are 25 000 people and the population increases by 5.5 %, or 0.055, annually. Then a= 25 000 and r= 0.055.
y= 25 000(1+ 0.055)^t
⇓
y=25 000(1.055)^t
This function represents the population y after t years.
b
We will start by finding a function that represents the population y after t months. To do so, we will use the fact that t= 112* 12t.
In this form of the function, the factor (1.0045) represents the monthly growth factor. Since the growth factor is equal to 1 plus the rate of growth, the monthly rate of growth r can be found.
1.0045 = (1+ r)
⇓
r = 0.0045 or 0.45 %
Therefore, the monthly percent increase is about 0.45 %.
c
To draw the graph of y= 25 000(1.055)^t we will make a table of values.
x
25 000(1.055)^t
y= 25 000(1.055)^t
1
25 000(1.055))^1
26 375
3
25 000(1.055)^3
≈ 29 356
5
25 000(1.055)^5
≈ 32 674
7
25 000(1.055)^7
≈ 36 367
The points ( 1, 26 375), ( 3, 29 356), ( 5, 32 674), and ( 7, 36 367) are on the graph of the function y= 25 000(1.055)^t.
Let's now plot and connect them with smooth curves.
We see that the population is about 30 000 after 4 years.
