Question

The population of a southern city follows the exponential law. Use this information to answer parts a and b.

(a) If n is the population of the city and t is the time in years, express n as a function of t.
n(t) = [Enter your response here]
(b) If the population doubled in size over 29 months, and the current population is 80,000, what will the population be 4 years from now?

Options:
Option 1: n(t) = 80,000e^(0.027t)
Option 2: n(t) = 40,000e^(0.054t)
Option 3: n(t) = 80,000e^(0.054t)
Option 4: n(t) = 40,000e^(0.027t)

Answer 1

(a) n(t) = 80,000e^(0.027t)

(b) The population will be approximately 106,137 four years from now.

Step-by-step explanation:

(a) The exponential growth formula is given by n(t) = n₀ * e^(kt), where n(t) is the population at time t, n₀ is the initial population, e is the base of the natural logarithm, and k is the growth rate.

In this case, the initial population (n₀) is 80,000, and the population doubled over 29 months. To find the growth rate (k), we use the formula for doubling time: 2 = e^(kt). Solving for k, we get k = ln(2)/29. Substituting this into the exponential growth formula, we get n(t) = 80,000e^(0.027t).

(b) To find the population four years from now (t = 4), we substitute t = 4 into the expression obtained in part (a). n(4) = 80,000e^(0.027*4) ≈ 106,137. Therefore, the population will be approximately 106,137 four years from now.

In summary, the population growth model for the southern city is n(t) = 80,000e^(0.027t), and the population is estimated to be around 106,137 four years from the current time.

This calculation is based on the given information and the exponential growth formula, providing a clear understanding of the population dynamics over time.