Respuesta :
Answer:
The rate at which the total personal income was rising in the Richmond-Petersburg area in 1999 is $1.627 billion per year
Step-by-step explanation:
Let [tex]t[/tex] be the number of years after 1999.
From the information given:
- In 1999, the population in this area was 961400, and the population was increasing at roughly 9200 people per year.
- The average annual income was 30593 dollars per capita, and this average was increasing at about 1400 dollars per year.
The population growth can be modeled with a linear equation. The initial population was [tex]P_0[/tex] is 961400 and it grows by 9200 people per year.
The population in time t can be written
[tex]P(t)=9200t+961400[/tex]
The average annual income can be modeled with a linear equation. The initial average annual income was 30593 dollars per capita and it grows by 1400 dollars per year.
[tex]A(t)=1400t+30593[/tex]
If we multiply both together gives the total personal income at time t.
[tex]T(t)=P(t)\cdot A(t)\\T(t)=(9200t+961400)\cdot (1400t+30593)[/tex]
The rate at which the total personal income was rising in the Richmond-Petersburg area is the derivative [tex]T(t)'[/tex]
We need to use the Product Rule that says
If f and g are both differentiable, then:
[tex]\frac{d}{dx}[f(x)g(x)]=f(x)\frac{d}{dx}[g(x)] +g(x)\frac{d}{dx}[f(x)][/tex]
Applying the Product Rule
[tex]\frac{d}{dt}T(t)=\frac{d}{dt} [(9200t+961400)\cdot (1400t+30593)]\\\\T(t)'=\frac{d}{dt}\left(9200t+961400\right)\left(1400t+30593\right)+\frac{d}{dt}\left(1400t+30593\right)\left(9200t+961400\right)\\\\T(t)'=9200\left(1400t+30593\right)+1400\left(9200t+961400\right)\\\\T(t)'=12880000t+281455600+12880000t+1345960000\\\\T(t)'=25760000t+1627415600[/tex]
For 1999, t = 0.
The raising is
[tex]T(0)'=25760000(0)+1627415600\\T(0)'=1,627,415,600[/tex]
The rate at which the total personal income was rising in the Richmond-Petersburg area in 1999 is $1.627 billion per year.
