A tank is being filled with water at a constant rate of 5 gallons per minute. After 3 minutes, the...

1. TRANSLATE the problem information

• Given information:
  • Filling rate: 5 gallons per minute (constant)
  • At \(\mathrm{t = 3}\) minutes: \(\mathrm{A = 40}\) gallons
  • Find: equation for A after t minutes
• What this tells us: We have a linear relationship where water increases at a steady rate.

2. INFER the approach

• Since the rate is constant, this follows the linear pattern: \(\mathrm{A = (rate \times time) + initial\ amount}\)
• We know the rate (5 gallons/minute) but need to find the initial amount
• Strategy: Use the 3-minute data point to work backwards and find what was in the tank at \(\mathrm{t = 0}\)

3. SIMPLIFY to find the initial amount

• Amount added in 3 minutes = \(\mathrm{5 \times 3 = 15}\) gallons
• Since tank contains 40 gallons after 3 minutes:
  Initial amount = \(\mathrm{40 - 15 = 25}\) gallons

4. TRANSLATE into final equation

• \(\mathrm{A = 5t + 25}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't realize they need to work backwards to find the initial amount. Instead, they might think that after 3 minutes, the tank contains exactly what was added (40 gallons), leading them to believe the equation should start from 0.

This reasoning leads them to think: "After 3 minutes we have 40 gallons, so the equation should be \(\mathrm{A = 5t}\), but that gives 15 gallons at \(\mathrm{t = 3}\), not 40. Let me add 40 to make it work: \(\mathrm{A = 5t + 40}\)."

This may lead them to select Choice B (\(\mathrm{A = 5t + 40}\)).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret what "after 3 minutes, the tank contains 40 gallons" means. They might think this means 40 gallons were added in 3 minutes, rather than understanding this is the total amount in the tank.

This leads them to set up: Initial amount + 40 = total after more time, causing confusion about the relationship between time, rate, and total amount.

This causes them to get stuck and guess among the remaining choices.

The Bottom Line:

The key insight is recognizing that "contains 40 gallons after 3 minutes" refers to the total amount, not just what was added. Students must work backwards using the rate to find the starting amount, which requires understanding the difference between "amount added" and "total amount."