In science class, Diego conducted an experiment to learn about evaporation. Diego measured the height of fluid in a beaker...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f(x) = 39 - 0.18x}\) gives estimated height in cm
  • \(\mathrm{x}\) = number of days after start of experiment
  • We need to interpret what 39 represents

2. INFER the mathematical structure

• This is a linear function in the form \(\mathrm{f(x) = b + mx}\)
• In our function: \(\mathrm{b = 39}\) and \(\mathrm{m = -0.18}\)
• The constant term (39) is the y-intercept

3. INFER what the y-intercept means in context

• The y-intercept occurs when \(\mathrm{x = 0}\)
• Since x represents "days after start of experiment," \(\mathrm{x = 0}\) means "at the start of the experiment"
• Therefore: \(\mathrm{f(0) = 39 - 0.18(0) = 39}\)

4. TRANSLATE back to context

• \(\mathrm{f(0) = 39}\) means the estimated height is 39 cm at the start of the experiment
• This matches exactly with choice A

Answer: A. The estimated height, in cm, of the fluid at the start of the experiment

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not clearly connect \(\mathrm{x = 0}\) to "the start of the experiment," or may confuse which parameter represents what aspect of the situation.

Without this connection, they might think 39 represents something else entirely, like the final height (Choice B) or misinterpret the role of the slope \(\mathrm{-0.18}\). This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Inadequate INFER reasoning: Students may recognize that 39 is a constant but fail to connect it to the y-intercept concept and what that means in the context.

They might focus on the more "active" coefficient \(\mathrm{-0.18}\) (which does represent change per day) and think the question is asking about that instead. This may lead them to select Choice C (\(\mathrm{-0.18}\) represents change per day) by confusing which number the question is asking about.

The Bottom Line:

This problem tests whether students can connect abstract mathematical concepts (y-intercept) to real-world meaning (starting condition). The key insight is recognizing that the constant term in a linear function represents the initial value when the input is zero.