Question

The populations P (in thousands) of Orlando, Florida from 2000 through 2007 can be modeled by

P = 1656.2e^kt

where t represents the year, with t =0 corresponding to 2000.

In 2005, the population of Orlando was about 1,940,000. (Source: U.S. Census Bureau)

(a) Find the value of k. Is the population increasing or decreasing? Explain.

(b) Use the model to find the populations of Orlando in 2010 and 2015. Are the results reasonable? Explain.

Answer 1

Explained below.

Explanation:

The population model for Orlando, Florida from 2000 through 2007 is:

`P = 1656.2* e^(kt)`

It is provided that, in 2005, the population of Orlando was about 1,940,000.

(a)

Compute the value of k as follows:

For year 2005, the value of t is 5.

`P = 1656.2* e^(kt)`

`1940000=1656.2* e^(k* 5)\\\\e^(k* 5)=(1940000)/(1656.2)\\\\e^(k* 5)=1171.35612\\\\5k=\ln(1171.35612)\\\\k=(7.06592)/(5)\\\\k=1.413184\\\\k\approx 1.41`

Since the value of k is positive, the population is increasing.

(b)

Compute the populations of Orlando in 2010 as follows:

`P = 1656.2* e^(kt)`

`=1656.2* e^(1.41* 10)\\\\=2201227729.68\\\\\approx 2,201,227,730`

Compute the populations of Orlando in 2015 as follows:

`P = 1656.2* e^(kt)`

`=1656.2* e^(1.41* 15)\\\\=2537704633019.96\\\\\approx 2,537,704,633,020`

Since the population increased exponentially from 2010 to 2015, yes the results are reasonable.