A water tank is leaking at a constant rate. The amount of water in the tank, \(\mathrm{W(t)}\), in gallons, is...

1. TRANSLATE the problem information

• Given information:
  • Water tank leaking at constant rate (linear relationship)
  • \(\mathrm{W(t)}\) represents gallons of water at time \(\mathrm{t}\) hours
  • At \(\mathrm{t = 0}\): tank has 150 gallons
  • At \(\mathrm{t = 4}\): tank has 138 gallons
• What this tells us: We have two coordinate points \(\mathrm{(0, 150)}\) and \(\mathrm{(4, 138)}\)

2. INFER the approach needed

• Since we need a linear function and have two points, we can use the slope-intercept form: \(\mathrm{W(t) = mt + b}\)
• Strategy: Find the slope (m) and y-intercept (b) to write the complete function

3. Find the y-intercept (b)

• The y-intercept occurs when \(\mathrm{t = 0}\)
• From the given information: \(\mathrm{W(0) = 150}\)
• Therefore: \(\mathrm{b = 150}\)

4. SIMPLIFY to find the slope (m)

• Use the slope formula with points \(\mathrm{(0, 150)}\) and \(\mathrm{(4, 138)}\):
• \(\mathrm{m = \frac{138 - 150}{4 - 0}}\)
• \(\mathrm{m = \frac{-12}{4} = -3}\)
• The negative slope makes sense: the tank is losing water over time

5. Write the complete function

• Substitute \(\mathrm{m = -3}\) and \(\mathrm{b = 150}\) into \(\mathrm{W(t) = mt + b}\)
• \(\mathrm{W(t) = -3t + 150}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may correctly identify the initial amount (150 gallons) but struggle to extract the second coordinate point from "After 4 hours, the tank contained 138 gallons." They might miss that this gives them the point \(\mathrm{(4, 138)}\).

Without both coordinate points, they cannot calculate the slope correctly and may guess or use incorrect values. This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students may correctly identify both points but make arithmetic errors when calculating the slope. For example, they might calculate \(\mathrm{\frac{138 - 150}{4 - 0} = \frac{-12}{4}}\) incorrectly, getting \(\mathrm{m = -12}\) instead of \(\mathrm{m = -3}\).

This may lead them to select Choice A (\(\mathrm{-12t + 150}\)) because they use the wrong slope value while keeping the correct y-intercept.

The Bottom Line:

This problem tests whether students can extract coordinate information from a word problem and then execute the mechanical process of finding a linear function. The key challenge is recognizing that real-world descriptions translate directly into mathematical coordinates.