Question

the table shows the population of a small town over time. the function $p = 10,550(1.1)^x$ models the population $x$ years after the year 2000.

population over time

years after 2000, $x$	population, $p$
3	14,000
4	15,500
7	20,500
10	27,000

for which year would this model most likely be sufficient to make a prediction of the population?

\\(\bigcirc\\) 1950
\\(\bigcirc\\) 2005
\\(\bigcirc\\) 2025
\\(\bigcirc\\) 2050

Explanation:

Step1: Analyze the model's domain

The function \(P = 10550(1.1)^x\) is defined for \(x\) years after 2000.

Step2: Check each option

• For 1950: \(x=1950 - 2000=- 50\), the model is for \(x\geq0\) (years after 2000), so 1950 is not valid.
• For 2005: \(x = 2005 - 2000=5\), within a reasonable range close to the data - points (data up to \(x = 10\)).
• For 2025: \(x=2025 - 2000 = 25\), while the model may still hold, it's a bit further from the data range (\(x = 0\) to \(x = 10\)) compared to 2005.
• For 2050: \(x=2050 - 2000=50\), very far from the data range (\(x = 0\) to \(x = 10\)) and many external factors (not accounted for in the simple exponential model) could affect the population over such a long - term.

Answer:

2005