We are given the population growth model:

$P(t) = 262,426e^{0.25t}$

where $P(t)$ is the population after $t$ years. We are asked to find $t$ such that the population reaches 1 million, i.e., $P(t) = 1,000,000$.

Step 1: Set up the equation

We want to solve for $t$ in the equation:

$1,000,000 = 262,426e^{0.25t}$

Step 2: Isolate the exponential term

Divide both sides of the equation by 262,426:

$\frac{1,000,000}{262,426} = e^{0.25t}$

$3.8101 \approx e^{0.25t}$

Step 3: Take the natural logarithm of both sides

To solve for $t$, take the natural logarithm ($\ln$) of both sides:

$\ln(3.8101) = 0.25t$

$1.338 \approx 0.25t$

Step 4: Solve for $t$

Now, solve for $t$:

$t = \frac{1.338}{0.25} = 5.352$

Step 5: Round the answer

Since we need to round to the nearest year, the number of years it will take for the population to reach 1 million is approximately 5 years.

Would you like further details or have any questions? Here are 5 related questions you might consider:

1. What would the population be after 10 years using the same model?
2. How does the growth rate affect the time to reach a certain population?
3. What is the population after 5 years based on this model?
4. Can this model be applied to predict when the population will reach 2 million?
5. What happens to the population if the growth rate is reduced by half?

Tip: Always check if exponential models accurately represent real-world growth, as assumptions may not hold long-term.