Question

The population p(in thousands) of colorado springs, colorado is given by p(t)=36e^kt where t represent the year, with t=0 corresponding to the year 2000. In 1980, the population was 215,000. Find the value of k

Answer 1

k ≈ 0.44

Explanation:

Given the equation

p(t) = 36e^(kt)

In the year 2000, when t = 0, the population

p(0) = 36e^k............(1)

In 1980, when t = -20, the population was 215000

This implies that

215000 = 36e^(-20k)

e^(-20k) = 215000/36

e^(-20k) = 5972.222

Taking natural logarithm of both sides

ln(e^(-20k)) = ln(5972.222)

-20k = 8.695

k = -8.695/20 = 0.435

k ≈ 0.44

Answer 2

Correction

The function is
`p(t)=361e^{kt`

Answer:

k=0.0259

Explanation:

The population function is given as:
`p(t)=361e^(kt)`

Where t=0 corresponds to the year 2000.

In 1980, 1980-2000=-20, p(-20)=215

Therefore:

`215=361e^(k*-20)\\$Divide both sides by 361$\\(215)/(361)= e^(-20k)\\$Take the natural logarithm of both sides$\\ln((215)/(361))=-20k\\k=ln((215)/(361))/ (-20)\\k=0.0259`