Question

A city has been declining in population for the last 50 years. In the year 2000, the population was 571 thousand, and was declining at a rate of 0.77% per year. If this trend continues: (a) Give a formula for the population of this city, P, in thousands, as a function of t, the number of years since 2000. P= (b) What is the predicted population in 2010? The predicted population is thousand people. (c) To two decimal places, estimate t when the population is 450 thousand people. When there are 450 thousand people in this city, t≈

Answer 1

what have you done so far?

Answer 2

Well, actually,P(t)=571000(.23)^t

Answer 3

\[\large 450=571e ^{0.0077t}\]

Answer 4

Solve for t.

Answer 5

I haven't learned about the #e yet.

Answer 6

Exponential is e.

Answer 7

Alright, well thank you for trying to help me!
I am just not allowed to use e in my equation because that is the wrong answer according to my teacher.

Answer 8

You don't necessarily have to use the constant 'e'.

Population = 571,000 * .9923^n where n= (present year minus 2000)

So, for an example, the population in 2003 would be:
571,000 * .9923^3 which equals
557,911

They want the formula to provide answers in thousands:
a)
So we have the formula:
Population (in thousands) = (571 * .9923^t)
where 't' equals the number of years since 2000

b)
The population in 2010 is
Population (in thousands) = (571 * .9923^10)
Population (in thousands) = 528.525592559
Population (in thousands) (rounded) = 528.526

c) To find when population will be 450,000 we have to manipulate the formula:
450 = (571 * .9923^x)

(450 / 571) = .9923^x

To find the time (or 'x') we must take logarithms of both sides:
log (450 / 571) = x * log (.9923)
log (0.7880910683) = x * log (.9923)
-0.1034235945 = x * -0.0033570086

x = -0.1034235945 / -0.0033570086

x = 30.8082597997 years or rounding this to 2 decimal places
x = 30.81 years

and see? It's just that simple - LOL

Answer 9

Haha. Thank you SO much! You are a lifesaver!

Answer 10

"wolframalpha" didn't get the formula quite right, so it's just as well that you couldn't use it :-)

If you set up the decay with the exponential function, the value of \(\lambda\) should be negative in order to get a positive value of \(t\).

\[450 = 591 e^{-0.0077298t}\]is what you would solve for \(t\).
\[\ln(\frac{450}{571}) = -0.0077298t\]\[-0.238142 = -0.0077298t\]\[t = 30.8083\]