Hey Guys I need helo with this math qustion

Question

City -Duluth

Population 588,450

Annual Percent Change (Growth Rate) - 3.2% dP/dt = 0.032P

1 - Find an exponential function of the form P(t)=P0e^kt , which gives the population of the county in the year 2000 + t. **(P(t)=P0e^kt) the 0 is lowered**
2 - Evaluate the function at t = 16 and give a verbal interpretation of your answer.
3 - Find the derivative of the function.
4 - Evaluate the derivative at t = 16 and give a verbal interpretation of your answer.

Answer 1

P₀ is the initial population, or population at time=0, P₀ = 588,450

you need to solve for k. k=growth constant. in other words, every whole number for k, is an increase of the number e.

 

        e¹ approx. = 2.718, e² approx. = 7.389, and so on. 

We need to equate this and the annual growth rate of .032 P.

 

 k=ln(1+r/100)/period. in your case the period is 1 year and r = 3.2

                 

                           k = ln(1+3.2/100)/1=0.0315

 

 therefore, P(t) = P₀ e^{0.0315 t}

 

 and P(16) = 588450 e^0.0315*16 = 974078.56, an increase of 385628 people.

 

taking the derivative,

 

        dp/dt = 588450*0.0315*e^{0.0315 t} = 18536.175 e^{0.0315 t}

 

evaluated at t=16,

 

          18536.175 e^0.0315*16 = 30683.47, the increased number of people in the 16th year.