Question:

The population of a certain species of bird is modeled by the function $P(t) = 500e^{0.05t}$, where $t$ is measured in years.

a) Calculate the population of the bird species after 10 years.

b) Determine the annual growth rate of the bird population.

c) Find the time it takes for the population to double.

Answer:

a) To find the population of the bird species after 10 years, we simply substitute $t = 10$ into the given function:

$P(10) = 500e^{0.05 \cdot 10} = 500e^{0.5} \approx 500 \cdot 1.6487 \approx 824.35$

So, the population after 10 years is approximately 824 birds.

b) The annual growth rate of the bird population is given by the constant in the exponent of the exponential function, which is $0.05$ in this case. Therefore, the annual growth rate is $5\%$ per year.

c) We need to find the time it takes for the population to double, so we set the population function equal to $2P_0$ (double the initial population):

Divide by 500:

Take the natural logarithm of both sides:

Solve for $t$:

So, it takes approximately 13.86 years for the population to double.

Therefore, after 10 years, the population of the bird species is approximately 824 birds, with an annual growth rate of 5%, and it takes approximately 13.86 years for the population to double.