Question

A town has a population of 4.11 x 10^4 and shrinks at a rate of 9.6% every year. Which equation represents the town's population after 4 years?

Answer 1

Let P(t) be the population of the town after t years. We know that P(0) = 4.11 x 10^4. Then, the rate of change of the population is given by -0.096P(t), so we can write the differential equation:

dP/dt = -0.096P(t)

To find the population after 4 years, we need to solve this differential equation with the initial condition P(0) = 4.11 x 10^4. We can do this by separating variables and integrating both sides:

∫(dP/P) = -0.096 ∫dt

ln|P(t)| = -0.096t + C

where C is a constant of integration that can be found using the initial condition. Solving for P(t), we get:

P(t) = Ce^(-0.096t)

where C = e^(ln|P(0)|) = e^(ln|4.11 x 10^4|) = 4.11 x 10^4

Substituting in t = 4, we get:

P(4) = 4.11 x 10^4 e^(-0.096 × 4) = 4.11 x 10^4 e^(-0.384)

So, the equation representing the town's population after 4 years is:

P(t) = 4.11 x 10^4 e^(-0.096t)

Answer 2

Explanation:

The equation that represents the town's population after 4 years can be found using the formula:

P = P0 * (1 - r)^t

Where P is the population after t years, P0 is the initial population (4.11 x 10^4), r is the annual shrink rate as a decimal (9.6% as 0.096), and t is the number of years (4).

P = 4.11 x 10^4 * (1 - 0.096)^4

P = 4.11 x 10^4 * 0.6994^4

P = 4.11 x 10^4 * 0.4139

P = 1.70 x 10^4

So the town's population after 4 years is 1.70 x 10^4.