A water tank is being drained at a constant rate. The relationship between the volume of water, V, in liters,...

1. TRANSLATE the problem information

• Given information:
  • Water tank draining at constant rate (linear relationship)
  • For every 1-minute increase, volume decreases by 15 liters
  • At \(\mathrm{t = 3}\) minutes, \(\mathrm{V = 455}\) liters
  • Need equation in the form \(\mathrm{V = mt + b}\)

• What this tells us: We have a linear relationship with negative slope and one known point.

2. INFER the approach

• Since it's linear, we need slope (m) and y-intercept (b)
• The rate of change gives us the slope directly
• The given point will help us find the y-intercept

3. TRANSLATE the rate information to find slope

• "For every increase in t by 1, V decreases by 15"
• This means: \(\mathrm{slope = -15}\) (negative because decreasing)
• Our equation becomes: \(\mathrm{V = -15t + b}\)

4. INFER how to find the y-intercept

• Use the known point \(\mathrm{(3, 455)}\) in our equation
• Substitute \(\mathrm{t = 3}\) and \(\mathrm{V = 455}\) into \(\mathrm{V = -15t + b}\)

5. SIMPLIFY to solve for b

• \(\mathrm{455 = -15(3) + b}\)
• \(\mathrm{455 = -45 + b}\)
• \(\mathrm{455 + 45 = b}\)
• \(\mathrm{b = 500}\)

6. Write the final equation

• \(\mathrm{V = -15t + 500}\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Misinterpreting "decreases by 15" as a positive slope instead of negative.

Students might think "by 15" means +15, leading them to write \(\mathrm{V = 15t + b}\). Using the point \(\mathrm{(3, 455)}\): \(\mathrm{455 = 15(3) + b}\) gives \(\mathrm{b = 410}\), producing \(\mathrm{V = 15t + 410}\).

This may lead them to select Choice D (\(\mathrm{V = 15t + 410}\)).

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic errors when solving for the y-intercept.

Students correctly identify \(\mathrm{slope = -15}\) but make calculation mistakes: \(\mathrm{455 = -15(3) + b}\) becomes \(\mathrm{455 = -45 + b}\), but they might calculate \(\mathrm{455 - 45 = 410}\) instead of \(\mathrm{455 + 45 = 500}\). This gives \(\mathrm{V = -15t + 410}\), but this option doesn't exist, leading to confusion and potentially selecting the similar-looking Choice A (\(\mathrm{V = -15t + 455}\)).

The Bottom Line:

This problem tests whether students can accurately translate rate language into mathematical slope and then execute the algebra correctly to find the complete linear equation.