\(\mathrm{P(t) = 250 + 10t}\) The population of snow leopards in a certain area can be modeled by the function...

1. TRANSLATE the given information

• Function: \(\mathrm{P(t) = 250 + 10t}\)
• \(\mathrm{P(t)}\) = population of snow leopards t years after 1990
• Need to interpret: \(\mathrm{P(30) = 550}\)

2. TRANSLATE what P(30) = 550 tells us

• The input is \(\mathrm{t = 30}\)
• The output is \(\mathrm{P(t) = 550}\)
• This means: when \(\mathrm{t = 30}\), the population P is 550

3. INFER what t = 30 means in context

• Since t represents "years after 1990"
• \(\mathrm{t = 30}\) means 30 years after 1990
• Calculate the actual year: \(\mathrm{1990 + 30 = 2020}\)

4. TRANSLATE the complete interpretation

• Population = 550 leopards
• Year = 2020
• Therefore: "The snow leopard population is predicted to be 550 in the year 2020"

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse which number represents what quantity.

They might think the "30" in \(\mathrm{P(30) = 550}\) represents the population instead of the time input. This leads them to interpret it as "population of 30 in some year" rather than "population in year corresponding to \(\mathrm{t = 30}\)."

This may lead them to select Choice A (population of 30 in 2020) or Choice B (population of 30 in 2030).

Second Most Common Error:

Poor INFER reasoning: Students incorrectly calculate the year.

They understand that \(\mathrm{t = 30}\) means "30 years after 1990" but miscalculate: \(\mathrm{1990 + 30 = 2030}\) instead of 2020. Sometimes they add the decades incorrectly or confuse which direction to add.

This may lead them to select Choice D (population of 550 in 2030).

The Bottom Line:

Function interpretation problems require careful attention to which number is the input and which is the output, plus accurate calculation of reference dates. The key is translating mathematical notation back to real-world context without mixing up the roles of different numbers.