Question

An initial population of 1,500 increases at an annual rate of 15%. Write an exponential function to model the population increase. Use the function to find the population after 15 years.

Answer 1

Answer:

An exponential growth function is in the form of $y =a(1+b)^x$ where a is the initial value and b is the growth rate and is not equal to 0, b> 1.

As per the given statement: An initial population of 1,500 increases at an annual rate of 15%.

let t  be the time in years.

Initial Population(a) = 1500

rate(b)  = 5%

then;

$(1+b)^t = (1+0.15)^t = (1.15)^t$ = increases 15% each year.

Then; by definition of exponential

we have an equation:

$P(t)= 1500(1.15)^t$ ......[1];

where 1500 represents the initial population.

Now, to find the population after 15 years.

t = 15 years

Substitute in equation [1] we get;

$P(15)= 1500(1.15)^{15}$

Simplify:

P(15) =12205.5924437 population.

Therefore, the population after 15 years is, 1,2205.59(nearest hundredths) population.