RaySix

Post

Exponential Growth and Population Projection

Created by @nathanedwards

at October 31st 2023, 2:18:49 pm.

AP_Calculus_AB-Questions

#population-growth

#exponential-function

#exponential-growth-rate

Question:

The population of a certain city is modeled by the function

P(t) = 5000 \cdot e^{0.02t}

where $P(t)$ represents the population at time $t$ in years.

a) Determine the population when $t = 10$ .

b) Determine the exponential growth rate of the population.

Answer:

a) To find the population when $t = 10$ , we need to evaluate $P(10)$ .

Given:

P(t) = 5000 \cdot e^{0.02t}

Substituting $t = 10$ :

P(10) = 5000 \cdot e^{0.02 \cdot 10}

Simplifying:

P(10) = 5000 \cdot e^{0.2}

Using a calculator, we evaluate $e^{0.2} \approx 1.221$ .

P(10) \approx 5000 \cdot 1.221 \approx 6105.5

Therefore, the population when $t = 10$ is approximately 6105.5.

b) The exponential growth rate refers to the constant multiplying the exponent of the exponential function. In this case, it is the coefficient of $t$ in the exponent.

From the given expression, the exponential growth rate is $0.02$ .

Therefore, the exponential growth rate of the population is $0.02$ .