A water tank is draining at a constant rate. The table below shows the linear relationship between the time in...

1. TRANSLATE the problem information

• Given information:
  • Table showing (time, volume) pairs: \(\mathrm{(10, 1050)}\), \(\mathrm{(20, 900)}\), \(\mathrm{(30, 750)}\)
  • Need equation in the form shown in answer choices
• What this tells us: We have a linear relationship where volume decreases as time increases

2. INFER the approach needed

• This is asking for a linear equation, so I need slope (m) and y-intercept (b)
• The answer choices are in rearranged form, so I'll need to manipulate my final equation

3. SIMPLIFY to find the slope

• Using any two points, I'll calculate: \(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)
• With \(\mathrm{(10, 1050)}\) and \(\mathrm{(20, 900)}\):
  \(\mathrm{m = \frac{900 - 1050}{20 - 10}}\)
  \(\mathrm{m = \frac{-150}{10}}\)
  \(\mathrm{m = -15}\)

4. SIMPLIFY to find the V-intercept

• Using \(\mathrm{V = mt + b}\) with point \(\mathrm{(10, 1050)}\):
  \(\mathrm{1050 = -15(10) + b}\)
  \(\mathrm{1050 = -150 + b}\)
  \(\mathrm{b = 1200}\)

5. INFER the complete equation and rearrange

• Linear equation: \(\mathrm{V = -15t + 1200}\)
• To match answer format, add 15t to both sides: \(\mathrm{15t + V = 1200}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when rearranging the equation or calculating the slope incorrectly.

For example, they might get the slope as +15 instead of -15 (forgetting that volume decreases over time), leading to \(\mathrm{V = 15t + b}\). When they try to match this to answer choices, they might select Choice C (\(\mathrm{15V + t = 1200}\)) by confusing which variable has the coefficient.

Second Most Common Error:

Poor TRANSLATE reasoning: Students might not clearly identify which variable is independent (t) versus dependent (V), or misread what form the final equation should take.

This confusion about variable roles can lead them to set up the relationship backwards or get lost trying to match their work to the answer choices, leading to guessing among the options.

The Bottom Line:

This problem tests whether students can systematically work through linear relationships while keeping track of variable relationships and algebraic manipulation. The key is methodically finding slope and intercept, then carefully rearranging to match the required form.