exponential equation

[Guest]

The population of Detroit has been declining since 1980. According to the 1980 census, this city's population was 1,200,000. By 1993, this population was 1 million.

a) Assuming that the decline is exponential, determine an equation that models Detroit's population since the year 1980. use x=0 for 1980. (Hint: there are two known points on the graph of this exponential: (0, 1,200,000) and (13, 100,000,000)).

b) If the decline has continued at the same rate, what is the population of Detroit predicted by this model in 2002, 22 years after the 1980 census?

 

[Guest]

The population of Detroit has been declining since 1980. According to the 1980 census, this city's population was 1,200,000. By 1993, this population was 1 million.

a) Assuming that the decline is exponential, determine an equation that models Detroit's population since the year 1980. use x=0 for 1980. (Hint: there are two known points on the graph of this exponential: (0, 1,200,000) and (13, 100,000,000)).

b) If the decline has continued at the same rate, what is the population of Detroit predicted by this model in 2002, 22 years after the 1980 census?

If you use a graphing calculator, the answer is y=3685109.97* .986073^x

b)y=3685109.97* .986073^22

2706804.35 which rounds to 2706804

 

[Guest]

thank you, how do you do that on the calculator?

 

[Guest]

If you have a texas instument graphing calculator

You go to STAT

then you go to EDIT

and then in L1, you plug in all the years

and on L2, you plug in the population

and then you go back to STAT

and click CALC

and then click on EXPREG

and click ENTER

and then you jus plug in all the information

 

[soroban]

Hello, angelasloan7038!

I'll do it the old-fashioned way . . .

The population of Detroit has been declining since 1980.
According to the 1980 census, this city's population was 1,200,000.
By 1993, this population was 1 million.

a) Assuming that the decline is exponential, determine an equation that models Detroit's population
since the year 1980. Use x=0 for 1980.
(Hint: there are two known points on the graph of this exponential: (0, 1,200,000) and (13, 100,000,000)).

b) If the decline has continued at the same rate, what is the population of Detroit
predicted by this model in 2002, 22 years after the 1980 census?

The general form of the function is: .P(t) .= .P_oe^-kt
. . where P_o is the initial population (at t = 0).

We are told that when t = 0, P = 1.2 (million)
. . We have: .1.2 .= .P_oe⁰ . ---> . P_o = 1.2
The function (so far) is: .P(t) .= .1.2e^-kt

We are told that when t = 13, P = 1 (million)
. . We have: .1 .= .1.2e^-13k
Solve for k: .e^-13k .= .1/1.2 .= .5/6
. . -13k .= .ln(5/6)
. . k .= ,(-1/13)ln(5/6) .≈ .0.014

(a) The population function is: .P(t) .= .1.2e^-0.014t (measured in millions)


(b) In 2002: . P(22) .= .1.2e^(-0.014)(22) .= .0.881898382 (million)

. . . Therefore, the predicted population is about 880,000.

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Sorry, aloha . . . you must have an error.
. . You have an <u>increase</u> to 2.7 million by 2002.