A water tank contains 144 gallons of water at time t = 0 minutes.The tank drains at a constant rate,...

1. TRANSLATE the problem information

• Given information:
  • Initial amount: 144 gallons at \(\mathrm{t = 0}\) minutes
  • Amount after 12 minutes: 18 gallons
  • Draining occurs at a constant rate
  • Need to find: amount after 10 minutes

2. INFER the mathematical approach

• Since the rate is constant, this creates a linear relationship
• We can use the linear function form: \(\mathrm{W(t) = mt + b}\)
• We have two data points: \(\mathrm{(0, 144)}\) and \(\mathrm{(12, 18)}\)
• Strategy: Use these points to find \(\mathrm{m}\) and \(\mathrm{b}\), then evaluate at \(\mathrm{t = 10}\)

3. TRANSLATE the initial condition

• At \(\mathrm{t = 0}\), \(\mathrm{W(0) = 144}\)
• This means: \(\mathrm{m(0) + b = 144}\), so \(\mathrm{b = 144}\)
• Our function becomes: \(\mathrm{W(t) = mt + 144}\)

4. SIMPLIFY to find the rate (slope)

• Use the second condition: \(\mathrm{W(12) = 18}\)
• Substitute: \(\mathrm{12m + 144 = 18}\)
• Solve for \(\mathrm{m}\):
  \(\mathrm{12m = 18 - 144 = -126}\)
• Therefore: \(\mathrm{m = -126/12 = -10.5}\) (use calculator)

5. INFER the complete function

• We now have: \(\mathrm{W(t) = -10.5t + 144}\)
• The negative slope makes sense: water is draining out

6. SIMPLIFY to find the answer

• Evaluate at \(\mathrm{t = 10}\):
  \(\mathrm{W(10) = -10.5(10) + 144}\)
• Calculate:
  \(\mathrm{W(10) = -105 + 144 = 39}\)
  (use calculator for \(\mathrm{-10.5 \times 10}\))

Answer: 39

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may misinterpret what the linear function represents or confuse which variable is independent vs dependent. Some students might think the rate is positive because water is 'flowing,' not recognizing that draining creates a negative rate.

This conceptual confusion about the direction of change can lead them to calculate \(\mathrm{m = +10.5}\) instead of \(\mathrm{m = -10.5}\), resulting in:
\(\mathrm{W(10) = 10.5(10) + 144 = 249}\) gallons.
This leads to confusion and guessing since this answer seems unreasonably large.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly set up the linear function but make arithmetic errors when solving \(\mathrm{12m + 144 = 18}\). They might calculate:
\(\mathrm{12m = 18 + 144 = 162}\)
instead of \(\mathrm{12m = 18 - 144 = -126}\), leading to \(\mathrm{m = 13.5}\).

This gives:
\(\mathrm{W(10) = 13.5(10) + 144 = 279}\) gallons,
which again seems unrealistic and causes them to get stuck and guess.

The Bottom Line:

This problem tests whether students can recognize that 'constant rate of draining' translates to a negative slope in a linear function, and whether they can accurately perform the arithmetic to find that slope. The key insight is that draining reduces the amount over time, so the rate of change must be negative.