A storage tank initially contains 300 gallons of water. The water is drained from the tank at a constant rate...

1. TRANSLATE the problem information

• Given information:
  • Initial amount of water: 300 gallons
  • Drainage rate: 15 gallons per minute
  • Variables: \(\mathrm{w}\) = water remaining (gallons), \(\mathrm{t}\) = time (minutes)

2. INFER the mathematical relationship

• Since water is being drained (removed), the amount remaining decreases over time
• This creates a linear relationship where:
  • Starting point (when \(\mathrm{t = 0}\)): \(\mathrm{w = 300}\)
  • Rate of decrease: 15 gallons per minute
• The equation structure is: \(\mathrm{w = (starting\, amount) - (amount\, drained)}\)

3. Build the equation step by step

• Amount drained after t minutes = \(\mathrm{15 \times t = 15t}\)
• Water remaining = Initial amount - Amount drained
• \(\mathrm{w = 300 - 15t}\)

4. Verify by checking the pattern

• At \(\mathrm{t = 0}\): \(\mathrm{w = 300 - 15(0) = 300}\) ✓ (matches initial condition)
• At \(\mathrm{t = 1}\): \(\mathrm{w = 300 - 15(1) = 285}\) ✓ (makes sense: 15 gallons drained)
• At \(\mathrm{t = 20}\): \(\mathrm{w = 300 - 15(20) = 0}\) ✓ (tank empty after 20 minutes)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students confuse "draining" with "filling" and think the water amount increases over time.

They might reason: "Water is being added at 15 gallons per minute, so \(\mathrm{w = 300 + 15t}\)." This fundamental misunderstanding of the situation's direction leads them to select Choice C (\(\mathrm{w = 300 + 15t}\)).

Second Most Common Error:

Poor TRANSLATE execution: Students correctly understand that water decreases but mix up which number represents the constant term versus the coefficient.

They might write something like \(\mathrm{w = 15t - 300}\) or \(\mathrm{w = 285t}\), either by incorrectly placing the 300 or by somehow combining the numbers incorrectly. This confusion about equation structure may lead them to select Choice A (\(\mathrm{w = 285t}\)) or Choice B (\(\mathrm{w = 15t - 300}\)).

The Bottom Line:

This problem tests whether students can correctly model a decreasing linear relationship. The key insight is recognizing that "draining" means subtraction, and the initial amount serves as the starting point from which we subtract the accumulated drainage.