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3. A town's population has been decreasing at a constant rate, In 2010, the population was 5,900 By 2012 the population jad dropped to 4,700 Assume the tren contiues

a. Predict te population in 2016
b. Identify the year in which the population will reach 0

4. A town has an initial population of 75,000. It grows at a constant rate of 2,500 per year for five year

a. 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 began

b. 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 months

b. Fine a reasonable domain and range for the function W

Respuesta :

carlosego

We have a problem where a population decreases per year, in order to make predictions we must first find the rate of decrease.

In 2010, the population was 5,900.

In the year 2012, the population had been reduced to 4700.

Then [tex]5900-4700=1200[/tex]

In two years the population was reduced in 1200.

Then you can say that the population is reduced by 600 people per year.

An equation that models the population according to the years is:

[tex]P=5900-600t[/tex]

Where t is the time in years and P is the population according to the years.

From 2010, until 2016, 6 years will have elapsed.

then t = 6

[tex]P =5900-600*6\\[/tex]

[tex]P=2300[/tex].

To know in which year the population will become zero, we make P = 0 and clear t

[tex]0=5900-600t[/tex]

[tex]t=9,833[/tex] years


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=75 000+2500t[/tex]

[tex]t=\frac{-75000}{2500}\\[/tex]

[tex]t=-30[/tex]


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

[tex]P=75,000[/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]100 000=75000+2500t[/tex]

[tex]25 000=2500t[/tex]

[tex]t =10 years[/tex].  



The third 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]

Ver imagen carlosego
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