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?

• 1950
• 2005
• 2025
• 2050

Explanation:

Step1: Analyze the model's domain

The function \(P = 10550(1.1)^x\) models the population \(x\) years after the year 2000. So \(x\) should be non - negative.
For 1950: \(x=1950 - 2000=- 50\) (not valid as \(x\) represents years after 2000).
For 2005: \(x = 2005 - 2000=5\).
For 2025: \(x=2025 - 2000 = 25\).
For 2050: \(x=2050 - 2000=50\).
Since the model is based on data from 2000 and after (as \(x\) in the table starts from 0 which is 2000), a value of \(x\) that is not too far from the data range in the table (where \(x\) values are \(0,3,4,7,10\)) is more reliable. A \(x = 25\) (2025) is more likely to be predicted using an exponential model \(P = 10550(1.1)^x\) compared to \(x = 50\) (2050) which is a much larger extrapolation.

Answer:

C. 2025