A tank's volume V, in liters, is modeled as a linear function of time t, in minutes. At t =...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{V(t)}\) is linear (so \(\mathrm{V(t) = mt + b}\))
  • At \(\mathrm{t = 3}\) minutes: \(\mathrm{V = 120}\) liters
  • At \(\mathrm{t = 9}\) minutes: \(\mathrm{V = 120}\) liters
• This gives us two coordinate points: \(\mathrm{(3, 120)}\) and \(\mathrm{(9, 120)}\)

2. INFER what the identical outputs tell us

• Notice both points have the same V-value (120)
• When a linear function has the same output for different inputs, the slope must be zero
• This means we're dealing with a horizontal line (constant function)

3. SIMPLIFY by calculating the slope

• Using slope formula: \(\mathrm{m = (V_2 - V_1)/(t_2 - t_1)}\)
• \(\mathrm{m = (120 - 120)/(9 - 3)}\)
  \(\mathrm{= 0/6}\)
  \(\mathrm{= 0}\)
• Slope = 0 confirms this is a constant function

4. INFER the complete function

• Since \(\mathrm{V(t) = mt + b}\) and \(\mathrm{m = 0}\): \(\mathrm{V(t) = 0 \cdot t + b = b}\)
• Using either point to find b: at \(\mathrm{t = 3}\), \(\mathrm{V = 120}\), so \(\mathrm{b = 120}\)
• Therefore: \(\mathrm{V(t) = 120}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students don't recognize that identical y-values immediately signal zero slope and constant function.

Instead, they might get confused by having a "function" that doesn't seem to depend on the input variable t. They may think something is wrong with their work or try to force a more complex relationship where none exists. This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor TRANSLATE execution: Students misread the coordinate pairs or mix up which variable is which.

For instance, they might try to use \(\mathrm{(120, 3)}\) and \(\mathrm{(120, 9)}\) as points instead of \(\mathrm{(3, 120)}\) and \(\mathrm{(9, 120)}\). This would give them a vertical line (undefined slope) rather than a horizontal line, leading to complete confusion about which answer choice to select.

The Bottom Line:

The key insight is recognizing that when a linear function produces identical outputs for different inputs, you have a constant function. Many students expect functions to "do something" with the input variable, so a function that ignores the input entirely feels counterintuitive.