The population of a city is modeled by the equation  P(t)=262,426e0.25t where t

is measured in years. If the city continues to grow at this rate, how many years will it take for the population to reach one million?

 

Round your answer to the nearest year.

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.

The population of a city is modeled by the equation P(t)=262,426e^0.25t where t is measured in years. If the city continues to grow at this rate, how many years will it take for the population to reach one million? Round your answer to the nearest year.

Discover how to solve for the time it takes a population to reach one million using an exponential growth model. This detailed solution covers key concepts like logarithms and exponential equations, suitable for high school students in Grades 10-12.

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