The city has an initial population of 75,000.
Grows at a constant rate of 2,500 per year for five years
a) We must find a linear function that models the P population of the city according to the years.
This function has the following form:
[tex]P=P_{0}+ at\\[/tex]
Where
P is the population as a function of time
[tex]P_{0}[/tex] is the initial population
"a" is the constant rate of growth of the function.
"t" is the time elapsed in units of years.
Then the function is:
[tex]P=75,000+2500t[/tex]
b) Before plotting the function, let's find its intercepts with the "t" and "P" axes
To find the intercept of the function with the t axis we do P = 0
[tex]0 =75000+2500t[/tex]
[tex]t=\frac{-75 000}{2500}[/tex]
[tex]t = -30[/tex]
Now we make t = 0 to find the intercept with the P axis
[tex]P =75000[/tex]
The intercept with the P axis at P = 75 000 means that this is the initial population, therefore, for a period of 0 to 5 years, the population can not be less than 75,000.
The intercept at t = -30 does not have an important significance for this problem, since we are evaluating population growth for a period of [tex]0 \leq t \leq 5[/tex].
The graph of the function is shown in the attached figure.
c) To answer this question we must do P = 100 000 and clear t.
[tex]100000=75000+2500t [/tex]
[tex]25 000=2500t [/tex]
[tex]t =10years[/tex].
The second problem is solved in the following way:
The weight of a newborn baby is 7.5 pounds
The baby earns half a pound a month in its first year
a) To find the function that models the weight of the baby we follow the same procedure as in the previous problem.
[tex]W = W_{0} + at[/tex]
Where
W is the baby's weight according to the months
[tex]W_{0}[/tex] is the initial weight in pounds
"a" is the rate of increase
"t" is the time elapsed in months.
So:
[tex]W = 7.5 + 0.5t[/tex]
b) The domain of the function is [tex]0 \leq t \leq 12\\[/tex]
Since the function only applies for the first year of growth of the baby, and one year has 12 months.
The range of the function is [tex]7.5 \leq W\leq 13.5[/tex]
