This exercise uses the population growth model.

Question

The population of a certain city was 102,000 in 2014, and the observed doubling time for the population is 17 years.

(a) Find an exponential model n(t) = n₀2^t/a

 for the population t years after 2014.

n(t) = 

 

(b) Find an exponential model n(t) = n₀e^rt

 for the population t years after 2014. (Round your r value to four decimal places.)

n(t) = 

 

Answer 1

The population of a certain city was 102,000 in 2014, and the observed doubling time for the population is 17 years.

(a) Find an exponential model n(t) = n₀2^t/a

 for the population t years after 2014.

First lets name our variables that are given to us:

n(t) = the population after t years

n₀ = 102,000 this is the initial population at time 0

t = the number of years that the population is growing for

a = 17 this is the number of years it takes for the population to double

After we know what the variables in our model stand for we can write the model:

n(t) = 102000(2)^t/17

(b) Find an exponential model n(t) = n₀e^rt

 for the population t years after 2014. (Round your r value to four decimal places.)

n(t) = 

We must first solve for the variable r:

We let n(t) = 2 and n₀ = 1, t = 17

We use 1 as our starting population and let it double to 2 over a period of 17 years. 

n(t) = n₀e^rt

2 = 1e^r(17)

ln(2) = lne^r(17)

ln(2) = r(17)

r = ln(2)/17

r = .0408

Now that we have the value of r we can write our new model:

n(t) = 102000e^.0408t