A population P satisfies dP/dt = 0.04P and has an initial population of 500. Find P(t) and determine the population after 10 years.

Step 1: Recognize the exponential growth model.

The differential equation is:
dP/dt = 0.04P

An equation of the form dP/dt = kP has the general solution:
P(t) = Ce^(kt)

Here:
k = 0.04

So:
P(t) = Ce^(0.04t)

Step 2: Apply the initial condition.

The initial population is 500, so:
P(0) = 500

Substitute t = 0:

500 = Ce^(0)

Since e^0 = 1:
C = 500

Step 3: Write the population model.

P(t) = 500e^(0.04t)

Step 4: Find the population after 10 years.

Substitute t = 10:

P(10) = 500e^(0.04 × 10)

P(10) = 500e^0.4

Step 5: Approximate the value.

e^0.4 ≈ 1.49182

Therefore:
P(10) ≈ 500(1.49182)
P(10) ≈ 745.91

Step 6: Interpret the result.

A population is normally expressed as a whole number, so the model predicts approximately 746 individuals after 10 years.

Therefore, P(t) = 500e^(0.04t), and P(10) ≈ 746.