Question

The population of a town (t) years after it was founded is given by p(t)=16000+400(t-5)A find P(5) and p(6)B what does -5 represent C what does 400(t-5) represent D write p(t) in slope intercept form

Answer 1

A) 16,000

B) 16,400

C) -5 represents that the population will grow after a little decrease

D) P(t) =400t +14000

1) Considering this function:

`P(t)=16000+400(t-5)`

A) Let's start by finding P(5) and P(6), i.e. the number of people when the instant of time (t) is 5 and 6.

So, let's plug that into the function:

`\begin{gathered} P(5)=16000+400(5-5) \\ P(5)=16,000+400(0) \\ P(5)\text{ =1}6,000 \end{gathered}`

For P(6)

`\begin{gathered} P(6)=16,000+400(6-5) \\ P(6)=16,000+400 \\ P(6)=16,400 \end{gathered}`

B) If we write the table, we can understand what doest -5 stands for

P(0)= 16,000+400(0-5)= 14000

P(1) = 16,000 +400(1-5)= 14,400

P(2) =16,000+ 400(2-5) = 14,800

(...)

The "t" number of years -5 shows that the population increases after having decreased in comparison to the first years.

C) The factor 400(t-5) represents the growth factor overtime of that population.

D) We need to rewrite that function into the Slope-intercept form:

`\begin{gathered} P(t)=16,000+400(t-5) \\ P(t)=16,000+400t-2000 \\ P(t)=400t+14000 \end{gathered}`

3) Hence, the answer is:

A) 16,000

B) 16,400

C) -5 represents that the population will grow after a little decrease

D) P(t) =400t +14000