What is the rate of change for the penguin population on the island at time t=0? (AP Calculus AB)

Question

The penguin population on an island is modeled by a differentiable function P of time t, where P(t) is the number of penguins and t is measured in years for 0≤t≤40. There are 100,000 penguins on the island at time t=0. The birth rate for the penguins on the island is modeled by

B(t)= 1000e^0.06tpenguins per year

and the death rate for the penguins on the island is modeled by

D(t)=250e^0.1tpenguins per year.

What is the rate of change for the penguin population on the island at time t=0?

Answer 1

The total population of penguins is the sum of the population and the birth rate minus the death rate

P(t) = 100000 + B(t) - D(t) = 100000 + 1000e^0.06t - 250e^0.1t

The rate of change of population is the derivative of this function.

P'(t) = 60e^0.06t - 25e^0.1t and evaluate at t = 0 to find the rate at that time.