Question

the population of a rural city follows the exponential growth model $p(t) = 3200e^{0.042t}$ where $t$ is the number of years after 1985
a) what was the population of this city in 1985?
b) what is the relative growth rate as a percent?
c) use this model to approximate the population in 2020.

a) the population in the year 1985 was \boxed{}.

Explanation:

Step1: Find t for 1985

t = 1985 - 1985 = 0

Step2: Calculate P(0)

$P(0) = 3200e^{0.042*0} = 3200e^0 = 3200*1 = 3200$

Step3: Identify relative growth rate

Exponential model: $P(t)=P_0e^{rt}$, so r=0.042=4.2%

Step4: Find t for 2020

t = 2020 - 1985 = 35

Step5: Calculate P(35)

$P(35)=3200e^{0.042*35}=3200e^{1.47}\approx3200*4.3499\approx13919.68\approx13920$

Answer:

a) 3200
b) 4.2%
c) 13920