The population size, in thousands, of a certain city can be modeled by the function P(t) = 268(1.02)^t, where t is the number of years after 2000 and t ≤ 10. Which of the following... The population size, in thousands, of a certain city can be modeled by the function P(t) = 268(1.02)^t, where t is the number of years after 2000 and t ≤ 10. Which of the following statements best describes the change in the population size in the years between 2000 and 2010? Read more

Steps to Solve

1. Identify Initial and Final Population

First, find the population at the start (2000) and end (2010) of the period using the function ( P(t) = 268(1.02)^t ).

For the year 2000 (( t = 0 )): $$ P(0) = 268(1.02)^0 = 268 \text{ (thousands)} $$

For the year 2010 (( t = 10 )): $$ P(10) = 268(1.02)^{10} $$

2. Calculate Final Population

Now, calculate ( P(10) ): Using ( 1.02^{10} ): $$ 1.02^{10} \approx 1.21899 $$

Thus, $$ P(10) = 268 \cdot 1.21899 \approx 326.152 \text{ (thousands)} $$

3. Find the Change in Population

To find the increase over the 10 years: $$ \text{Increase} = P(10) - P(0) $$ $$ \text{Increase} = 326.152 - 268 = 58.152 \text{ (thousands)} $$

4. Convert to Actual Numbers

Convert the increase to actual population: $$ \text{Increase in people} = 58.152 \times 1000 \approx 58152 \text{ people} $$

5. Analyze Statements

Now, review the original statements:

• A: Incorrect, as it suggests an increase of only 2%.
• B: Incorrect, as it suggests a 10% increase over 1 year instead.
• C: Incorrect, as it states an increase of 20,000, which does not match.
• D: Correct, as it indicates an increase of approximately 53,600, which is most accurate when rounded.