A water tank is filled at a constant rate. After 2 minutes, the tank contains 14 liters of water, and...

1. TRANSLATE the problem information

• Given information:
  • At \(\mathrm{t = 2}\) minutes: \(\mathrm{y = 14}\) liters
  • At \(\mathrm{t = 5}\) minutes: \(\mathrm{y = 26}\) liters
  • Tank fills at constant rate (linear relationship)
  • Need to find: \(\mathrm{y}\) when \(\mathrm{t = 0}\)

• What this tells us: We have two coordinate pairs \(\mathrm{(2, 14)}\) and \(\mathrm{(5, 26)}\), and we need the y-intercept

2. INFER the approach

• Since the relationship is linear, we can use \(\mathrm{y = mt + b}\)
• Strategy: Find the slope \(\mathrm{m}\) first using the two points, then substitute back to find \(\mathrm{b}\)
• The value \(\mathrm{y(0) = b}\) is exactly what we're looking for

3. SIMPLIFY to find the slope

• Using slope formula: \(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)
• \(\mathrm{m = \frac{26 - 14}{5 - 2}}\)
  \(\mathrm{m = \frac{12}{3}}\)
  \(\mathrm{m = 4}\) liters per minute

4. SIMPLIFY to find the y-intercept

• Use the linear equation \(\mathrm{y = mx + b}\) with point \(\mathrm{(2, 14)}\):
• \(\mathrm{14 = 4(2) + b}\)
  \(\mathrm{14 = 8 + b}\)
  \(\mathrm{b = 6}\)

Answer: B (6)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students often misunderstand what \(\mathrm{y(0)}\) represents or fail to properly extract the coordinate pairs from the word problem.

Some students think they need to find the rate at \(\mathrm{t = 0}\) rather than the amount at \(\mathrm{t = 0}\). Others might try to use the given values directly without recognizing they represent points on a line. This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when calculating the slope or solving for the y-intercept.

Common calculation mistakes include getting \(\mathrm{\frac{12}{3} = 3}\) instead of 4, or errors like \(\mathrm{14 = 8 + b}\) leading to \(\mathrm{b = 8}\) instead of \(\mathrm{b = 6}\). This may lead them to select Choice C (8) or other incorrect values.

The Bottom Line:

This problem tests whether students can translate a real-world linear relationship into mathematical form and systematically find the y-intercept. The key insight is recognizing that "\(\mathrm{y}\) when \(\mathrm{t = 0}\)" is asking for the y-intercept of the linear function, not some special calculation.