The volume of water in a leaking tank is a linear function of time. When t = 6 minutes, the...

1. TRANSLATE the problem information

• Given information:
  • Volume is a linear function of time
  • At t = 6 minutes: v = 32 liters
  • At t = 8 minutes: v = 25 liters
• What this tells us: We have two coordinate points (6, 32) and (8, 25) and need to find the equation \(\mathrm{v = mt + b}\)

2. INFER the approach

• Since we have two points on a line, we can find the slope first
• Then use one point with the slope to find the y-intercept
• This gives us the complete linear equation

3. SIMPLIFY to find the slope

• Using slope formula: \(\mathrm{m = \frac{v_2 - v_1}{t_2 - t_1}}\)
• \(\mathrm{m = \frac{25 - 32}{8 - 6} = -\frac{7}{2}}\)
• The negative slope makes sense - the tank is leaking, so volume decreases over time

4. SIMPLIFY to find the y-intercept

• Use the equation \(\mathrm{v = -\frac{7}{2}t + b}\) with point (6, 32):
• \(\mathrm{32 = -\frac{7}{2}(6) + b}\)
• \(\mathrm{32 = -21 + b}\)
• \(\mathrm{b = 53}\)

5. APPLY CONSTRAINTS to verify our answer

• Our equation: \(\mathrm{v = -\frac{7}{2}t + 53}\)
• Check with second point (8, 25): \(\mathrm{-\frac{7}{2}(8) + 53 = -28 + 53 = 25}\) ✓
• This matches our given information

Answer: D) \(\mathrm{v = -\frac{7}{2}t + 53}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Making sign errors when calculating the slope or solving for the y-intercept

Students might calculate the slope as \(\mathrm{+\frac{7}{2}}\) instead of \(\mathrm{-\frac{7}{2}}\), either by reversing the subtraction order or dropping the negative sign. This leads them to find a positive y-intercept and select Choice B (\(\mathrm{v = \frac{7}{2}t + 11}\)).

Second Most Common Error:

Poor TRANSLATE reasoning: Confusing which variable represents which quantity or misinterpreting the coordinate pairs

Students might mix up the time and volume values, or incorrectly set up their coordinate points. This leads to calculation errors throughout and may cause them to select Choice A or C depending on their specific mistakes.

The Bottom Line:

This problem requires careful attention to signs and systematic application of the slope-intercept method. The negative slope is crucial since it represents a leaking tank, and students must maintain accuracy through multiple algebraic steps.