A water tank is drained at a constant rate. After 2 hours, the tank contains 680 liters of water, and...

1. TRANSLATE the problem information

• Given information:
  • Tank drains at constant rate → linear function \(\mathrm{W(t) = mt + b}\)
  • After 2 hours: 680 liters → point \(\mathrm{(2, 680)}\)
  • After 5 hours: 500 liters → point \(\mathrm{(5, 500)}\)
• What this tells us: We have two points on a line and need to find the y-intercept b

2. INFER the solution strategy

• Since we have two points and need to find b in \(\mathrm{W(t) = mt + b}\), we must:
  • First find the slope m using the two points
  • Then substitute one point into \(\mathrm{W(t) = mt + b}\) to solve for b

3. SIMPLIFY to find the slope

• Using slope formula: \(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)
• \(\mathrm{m = \frac{500 - 680}{5 - 2}}\)
  \(\mathrm{= \frac{-180}{3}}\)
  \(\mathrm{= -60}\)
• So our function is \(\mathrm{W(t) = -60t + b}\)

4. SIMPLIFY to find b using point substitution

• Using point \(\mathrm{(2, 680)}\): \(\mathrm{680 = -60(2) + b}\)
• \(\mathrm{680 = -120 + b}\)
• \(\mathrm{b = 680 + 120 = 800}\)

5. Verify with the second point

• \(\mathrm{W(5) = -60(5) + 800}\)
  \(\mathrm{= -300 + 800}\)
  \(\mathrm{= 500}\) ✓

Answer: (E) 800

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students might confuse which variable represents what, or misinterpret the given information as something other than coordinate points.

For example, they might think b should equal one of the given water amounts (680 or 500) without understanding that b represents the y-intercept (initial amount when \(\mathrm{t = 0}\)). This could lead them to select Choice (A) (500) or Choice (C) (680).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly understand the approach but make arithmetic errors when calculating the slope or solving for b.

A common mistake is getting the slope calculation wrong (perhaps computing \(\mathrm{180/3 = 60}\) instead of \(\mathrm{-180/3 = -60}\)) or making sign errors during substitution. This leads to an incorrect value of b and may cause them to select Choice (D) (740) or get confused and guess.

The Bottom Line:

This problem tests whether students can work systematically with linear functions from real-world contexts, requiring careful translation of word problems into mathematical relationships and methodical algebraic manipulation.