A bird population is initially 150 and grows at a rate of 6% each year.

Since these populations grow exponentially, we are looking for a model of the form

P(t) = C*a^t, t is the time, C and a are the constants that we need to find

a. Find an exponential model that models the bird population P, after t years.

A bird population is initially 150 and grows at a rate of 6% each year.

This means that for t = 0 we have P(0) = 150 thus C*a^0 = C and C = 150

A bird population grows at a rate of 6% each year:

(P(1) - P(0))/P(0) = 0.06

P(1) - P(0) = 0.06*P(0)

150a - 150 = 150*0.06

a = (150*0.06 + 150)/150

a = 1.06

now we can write for the model of the bird population

P(t) = 150*1.06^t

b. Find the instantaneous growth rate r, as a percent. Round your answer to 5 decimal places.

To find the instantaneous growth rate differentiate the expression P(t) = 150*1.06^t

dP(t)/dt = 150*1.06^t*(ln 1.06)

r = 150*1.06^t*(ln 1.06)

r = 8.7403362*1.06^t

c. Write an exponential equation of the form P = Ce^(rt) for the bird population P after t years.

P(t) = Ce^(rt)

P(0) = 150 => C = 150

(P(1) - P(0))/P(0) = 0.06

150(e^r - 1)/150 = 0.06

e^r - 1 = 0.06

r = ln 1.06

r = 0.05827

thus the equation is

P(t) = 150 e^( 0.05827t).