Question

Suppose a town has 3500 people living in it, and ten years later, the town has a population of 1500. What is the growth constant, k, for this problem? How many people live in the town after 20 years?

A. k=−0.094, 1700 people
B. k=−0.029, 1000 people
C.k=0.029, 900 people
D.k=0.094, 700 people

Answer 1

The growth constant k for the town's population is -0.094, determined using the formula P(t) = P0e^(kt). Using this constant, the town's population after 20 years is calculated to be approximately 700 people, making Option D the correct answer.

Step-by-step explanation:

To determine the growth constant k for the town's population, we can use the continuous exponential growth formula P(t) = P0ekt, where P(t) is the population at time t, P0 is the initial population, and e is the base of the natural logarithm. Initially, the town has 3500 people, and after 10 years, it has 1500 people. We need to solve for k in the equation:

1500 = 3500e10k.

Dividing both sides by 3500, we get:

e10k = 1500/3500 = 0.4286.

Taking the natural logarithm of both sides gives:

10k = ln(0.4286),

and then we divide both sides by 10:

k = ln(0.4286)/10 ≈ -0.094.

To calculate the population after 20 years, we can apply the same formula with t = 20:

P(20) = 3500e(-0.094)(20) = 3500e-1.88 ≈ 700.

Therefore, Option D is correct with k = -0.094, and after 20 years, the population would be approximately 700 people.