Question

The population of Collin County, which follows the exponential growth model,

Answer 1

Answer:

(a) k=0.06217

(b) $P=264036e^{0.06217t}$

(c) 556,755 is the population in 2012

Step-by-step explanation:

Population increased from 264,036 in 1990 to 491,675 in 2000.

Let assume , year 1990 as t=0

When t=0, population P = 264,036

When t=10, population P = 491,675

Exponential growth equation is P=P_0e^(kt)

$P_0$ is the initial population when t=0

So $P_0= 264,036$

When t=10, population P = 491,675

Plug in all the values and find out k

$491675 = 264036 e^{k*10}$

Divide both sides by 264036

$\frac{491675}{264036}=e^{k*10}$

To remove 'e' we take ln on both sides

$ln(\frac{491675}{264036})=lne^{10k}$

$ln(\frac{491675}{264036})=10kln(e)$

The value of ln(e) = 1

$ln(\frac{491675}{264036})=10k$

Divide both sides by 10

k=0.06217

Exponential growth equation is $P=P_0e^{kt}$

We know  $P_0= 264,036$

Replace the value of k

$P=264036e^{0.06217t}$

Now , we find the population in 2012

For year 2012 , t= 12

So we plug on 12 for t  and solve for P

$P=264036e^{0.06217*12}$

P = 556755.09504

556,755 is the population in 2012