AP Calculus AB Exam Question

The population of a town is modeled by the function $P(t) = 5000e^{0.02t}$, where $P(t)$ represents the population in thousands at time $t$ in years.

a) Find the population of the town after 5 years.

b) Determine the annual growth rate of the population.

c) Estimate how long it will take for the population to double.

d) Find the population after 20 years and determine the annual growth rate at that time.

Answer

a) To find the population after 5 years, we substitute $t = 5$ into the equation:

$P(5) = 5000e^{0.02(5)}$

$P(5) = 5000e^{0.1}$

Using a calculator, we find $P(5) \approx 5997.51$.

The population of the town after 5 years is approximately 5,997.51 thousands.

b) The annual growth rate is given by the exponent of the exponential function. In this case, the annual growth rate is $0.02$, or 2%.

c) We want to find the time it takes for the population to double. Let $P_{\text{double}}$ represent the population of the town when it doubles.

$P_{\text{double}} = 5000e^{0.02t}$

$2 \cdot 5000 = 5000e^{0.02t}$

Dividing both sides of the equation by 5000, we have:

$2 = e^{0.02t}$

Taking the natural logarithm (ln) of both sides to solve for $t$:

$\ln(2) = 0.02t$

$t = \frac{\ln(2)}{0.02} \approx 34.66$

The population will take approximately 34.66 years to double.

d) To find the population after 20 years, we substitute $t = 20$ into the equation:

$P(20) = 5000e^{0.02(20)}$

$P(20) = 5000e^{0.4}$

Using a calculator, we find $P(20) \approx 7323.26$.

The population of the town after 20 years is approximately 7,323.26 thousands.

The annual growth rate at that time is still $0.02$, or 2%.