\(\mathrm{P(y) = 10,000(0.8)^{(y - 2020)}}\)The function P models the estimated population of a certain town in year y, where y...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{P(y) = 10,000(0.8)^{(y - 2020)}}\) models population in year y
  • We need the population in year 2020
• What this tells us: We need to evaluate \(\mathrm{P(y)}\) when \(\mathrm{y = 2020}\)

2. SIMPLIFY by substituting the value

• Substitute \(\mathrm{y = 2020}\) into the function:
  \(\mathrm{P(2020) = 10,000(0.8)^{(2020 - 2020)}}\)

3. SIMPLIFY the exponent

• Calculate the exponent: \(\mathrm{2020 - 2020 = 0}\)
• The expression becomes: \(\mathrm{P(2020) = 10,000(0.8)^0}\)

4. SIMPLIFY using the zero exponent rule

• Apply the rule that any non-zero number raised to power 0 equals 1:
  \(\mathrm{(0.8)^0 = 1}\)
• Calculate: \(\mathrm{P(2020) = 10,000 \times 1 = 10,000}\)

Answer: C (10,000)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students forget or don't know that any non-zero number raised to the power 0 equals 1.

Instead, they might think \(\mathrm{(0.8)^0 = 0.8}\) (treating the exponent as if it doesn't change the base), leading to \(\mathrm{P(2020) = 10,000 \times 0.8 = 8,000}\).

This may lead them to select Choice B (8,000).

Second Most Common Error:

Poor TRANSLATE reasoning: Students might misunderstand what the variable y represents or get confused about which year to substitute.

They might substitute a different value or get lost in the setup, leading to confusion and guessing.

The Bottom Line:

This problem tests whether students can execute basic function evaluation and remember fundamental exponent rules. The zero exponent rule is the key concept that trips up many students, since it's counterintuitive that raising a number to the power 0 always gives 1.