4. a town has an initial population of 75,000. it grows at a constant rate of 2,500 per year for five yearsa. find the linear function that models the town's population p as a function of the year, t where t is the number of years since the model beganb. graph y==p(t) interpret the meaning of the intercepts. c. when does the model expect the population to reach 100,000? 5. the weight of a newborn baby is 7.5 pounds. the baby gained one-half pound a month in its first year. a. find the linear function that models the baby's weight w as a function of the age of the baby t, in monthsb. fine a reasonable domain and range for the function w

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:

Where

P is the population as a function of time

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:

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

Now we make t = 0 to find the intercept with the P axis

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 .

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.

.

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.

Where

W is the baby's weight according to the months

is the initial weight in pounds

"a" is the rate of increase

"t" is the time elapsed in months.

So:

b) The domain of the function is

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