In the 2000 U.S.​ Census, a small city had a population of 40 comma 000.  By the​ 2010, the population had reached 59 comma 210.  If the population grows by the same percent each​ year, when will the population reach​ 100,000?

To solve this problem, we can use the formula for exponential growth:

\[
P(t) = P_0 \cdot (1 + r)^t
\]

Where:
- \(P(t)\) is the population at time \(t\),
- \(P_0\) is the initial population,
- \(r\) is the growth rate per year, and
- \(t\) is the number of years.

### Step 1: Find the annual growth rate \(r\)

We know that in 2000, the population was 40,000, and in 2010, the population was 59,210. So, over 10 years, the population grew from 40,000 to 59,210. We can use the formula:

\[
P(t) = P_0 \cdot (1 + r)^t
\]

Substitute the values:
\[
59,210 = 40,000 \cdot (1 + r)^{10}
\]

Solve for \(r\):

\[
\frac{59,210}{40,000} = (1 + r)^{10}
\]

\[
1.48025 = (1 + r)^{10}
\]

Now, take the 10th root of both sides to solve for \(r\):

\[
1 + r = 1.48025^{1/10}
\]

\[
1 + r \approx 1.04014
\]

\[
r \approx 0.04014
\]

So, the annual growth rate is approximately 4.014%.

### Step 2: Find when the population reaches 100,000

Now, we can use the same formula to find \(t\) when the population reaches 100,000:

\[
100,000 = 40,000 \cdot (1 + 0.04014)^t
\]

Divide both sides by 40,000:

\[
2.5 = (1.04014)^t
\]

Take the natural logarithm of both sides:

\[
\ln(2.5) = t \cdot \ln(1.04014)
\]

Now solve for \(t\):

\[
t = \frac{\ln(2.5)}{\ln(1.04014)}
\]

\[
t \approx \frac{0.9163}{0.03937}
\]

\[
t \approx 23.28
\]

So, it will take approximately 23.28 years from the year 2000 for the population to reach 100,000. This means it will happen around the year:

\[
2000 + 23.28 \approx 2023
\]

Thus, the population will reach 100,000 around the year **2023**.

Would you like further clarification on any of these steps?

---

Here are 5 related questions for further practice:
1. How would the growth rate change if the population reached 60,000 in 2010 instead?
2. What would the population be in 2030 if the same growth rate continues?
3. If the growth rate increased to 5%, when would the population reach 100,000?
4. How would you solve for the growth rate if only given the years and populations at two different times?
5. What if the population in 2020 was only 80,000—how would that affect the projection?

**Tip:** When dealing with exponential growth, remember to always check if the growth rate is expressed as a percentage or a decimal!

In the 2000 U.S. Census, a small city had a population of 40,000. By 2010, the population had reached 59,210. If the population grows by the same percent each year, when will the population reach 100,000?

Explore how to calculate when a population grows from 40,000 to 100,000 using exponential growth formulas. Learn how to apply logarithms and percentage growth to real-world scenarios, with a detailed step-by-step solution. Suitable for students in Grades 9-12.

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