Question

The town's population increases from 15,000 to 40,000 over a 10-year period. The Town Council expects the population to continue to grow and would like a mathematical description that can be used to predict future population size. Construct an exponential model for the town's population growth.

a. Exponential model: P(t) = 15,000 * (1 + r)^t
b. Exponential model: P(t) = 40,000 * e^(kt)
c. Exponential model: P(t) = 15,000 * e^(kt)
d. Exponential model: P(t) = 40,000 * (1 + r)^t

Give the growth rate for this town's population, and explain what the growth rate means in the context of this problem.

a. Growth rate: r = 0.1; It represents a 10% increase per year.
b. Growth rate: k = 0.2; It represents a 20% increase per year.
c. Growth rate: r = 0.2; It represents a 20% increase per year.
d. Growth rate: k = 0.1; It represents a 10% increase per year.

Approximate when the town's population will reach 100,000.

a. Around 15 years
b. Around 20 years
c. Around 25 years
d. Around 30 years

Answer 1

The exponential model for the town's population growth is P(t) = 15,000 * (1 + 0.147)^t. The town's population will reach 100,000 approximately 26.13 years after the initial 10-year period.

Step-by-step explanation:

The town's population is experiencing exponential growth, which means that the population is increasing at a constant percentage rate over time. To construct an exponential model for the town's population growth, we can use the formula P(t) = 15,000 * (1 + r)^t, where P(t) represents the population at time t, and r represents the growth rate. Since the population increased from 15,000 to 40,000 over a 10-year period, we can calculate the growth rate using the formula r = (P1/P0)^(1/t) - 1, where P1 is the final population, P0 is the initial population, and t is the number of years. Plugging in the values, we get r = (40,000/15,000)^(1/10) - 1 ≈ 0.147. Therefore, the exponential model for the town's population growth is P(t) = 15,000 * (1 + 0.147)^t.

Now, to approximate when the town's population will reach 100,000, we can set P(t) = 100,000 and solve for t. 100,000 = 15,000 * (1 + 0.147)^t. Dividing both sides by 15,000, we get (1 + 0.147)^t ≈ 6.67. Taking the natural logarithm of both sides, t * ln(1 + 0.147) ≈ ln(6.67). Finally, solving for t, we get t ≈ ln(6.67) / ln(1 + 0.147). Using a calculator, we find that t ≈ 16.13. Therefore, the town's population will reach 100,000 approximately 16.13 years after the initial 10-year period, totaling around 26.13 years.