\(\mathrm{T(t) = -t^2 + 12t + 28}\)The quadratic function above models the temperature above the freezing point T, in degrees...

1. TRANSLATE the question requirements

• We need to find the real-life meaning of the positive x-intercept
• Since \(\mathrm{T(t)}\) represents temperature above freezing point, we need to find when \(\mathrm{T(t) = 0}\)
• The x-intercept occurs when the function equals zero

2. INFER what \(\mathrm{T(t) = 0}\) means physically

• When \(\mathrm{T(t) = 0}\), the temperature above freezing point is 0 degrees
• This means the liquid is exactly at the freezing point
• So we need to solve: \(\mathrm{-t^2 + 12t + 28 = 0}\)

3. SIMPLIFY using the quadratic formula

Set up the quadratic formula with \(\mathrm{a = -1, b = 12, c = 28}\):

\(\mathrm{t = \frac{-12 ± \sqrt{144 + 112}}{-2}}\)

\(\mathrm{t = \frac{-12 ± \sqrt{256}}{-2}}\)

\(\mathrm{t = \frac{-12 ± 16}{-2}}\)

This gives us: \(\mathrm{t = -2}\) or \(\mathrm{t = 14}\)

4. APPLY CONSTRAINTS to select the meaningful solution

• The positive x-intercept is \(\mathrm{t = 14}\) minutes
• The negative solution (\(\mathrm{t = -2}\)) has no physical meaning since it represents a time before the liquid was removed from the heat source

5. INFER the real-world interpretation

• At \(\mathrm{t = 14}\) minutes, \(\mathrm{T(t) = 0}\), meaning the liquid reaches the freezing point
• This matches choice (D): "The time at which the liquid reaches the freezing point"

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skills: Students misunderstand what "temperature above freezing point" means and don't realize that \(\mathrm{T(t) = 0}\) corresponds to the liquid being at freezing point, not below freezing or at some other special temperature.

They might think \(\mathrm{T(t) = 0}\) means the liquid has stopped cooling or reached some minimum temperature other than freezing point. This conceptual misunderstanding leads them to select Choice (B) (maximum temperature) or causes confusion and guessing.

Second Most Common Error:

Poor INFER reasoning: Students correctly find the x-intercepts but don't connect the mathematical result to the physical meaning. They may focus on the mathematical properties (like maximum value occurs at the vertex) rather than interpreting what \(\mathrm{T(t) = 0}\) specifically means in this context.

This may lead them to select Choice (C) (time at maximum temperature) because they're thinking about vertex properties rather than x-intercept meaning.

The Bottom Line:

This problem challenges students to bridge mathematical concepts (x-intercepts) with real-world interpretation, requiring them to understand both what makes a function equal zero and what that means in the specific context of temperature above freezing point.