Math Snap

STEP 1

1. The initial population of Springfield in 2005 is 15,000.
2. The population increases by 4% each year for the next 5 years.
3. We need to find an equation that models this population growth.
4. The variable $x$ represents the number of years after 2005, where $x \leq 5$.

STEP 2

1. Understand the concept of exponential growth.
2. Determine the growth factor.
3. Write the exponential growth equation.
4. Compare with given options.

STEP 3

Understand the concept of exponential growth.
In exponential growth, the population increases by a constant percentage each year. This means the population for each subsequent year is a fixed multiple of the previous year's population.

STEP 4

Determine the growth factor.
Since the population increases by 4% each year, the growth factor is $1 + 0.04 = 1.04$.

STEP 5

Write the exponential growth equation.
The general form of an exponential growth equation is $p = p_0 \cdot (1 + r)^x$, where $p_0$ is the initial population, $r$ is the growth rate, and $x$ is the number of years.
For this problem, $p_0 = 15,000$, $r = 0.04$, and the growth factor is $1.04$.
Thus, the equation is:
$$p = 15,000 \cdot (1.04)^x$$

Compare with given options.
The equation $p = 15,000 \cdot (1.04)^x$ matches option (D).
The correct equation is:
$$\boxed{p = 15,000 \cdot (1.04)^x}$$