A water tank initially contains 1,250 gallons of water. Each day, 35 gallons of water are removed from the tank....

1. TRANSLATE the problem information

• Given information:
  • Initial water amount: 1,250 gallons
  • Daily removal: 35 gallons (constant amount)
  • No refilling occurs
• What this tells us: We need to find what type of function models water amount over time

2. INFER the mathematical relationship

• Since the same amount (35 gallons) is removed each day, we have a constant rate of change
• This suggests we're dealing with a linear relationship, not exponential
• Let's build the pattern to confirm

3. Build the function step by step

• After 1 day: \(1,250 - 35\) gallons
• After 2 days: \(1,250 - 35 - 35 = 1,250 - 2(35)\) gallons
• After 3 days: \(1,250 - 3(35)\) gallons
• After t days: \(\mathrm{W(t)} = 1,250 - 35\mathrm{t}\) gallons

4. INFER the function type and direction

• This has the form \(\mathrm{y} = \mathrm{mx} + \mathrm{b}\) where \(\mathrm{m} = -35\) and \(\mathrm{b} = 1,250\)
• This confirms it's a linear function
• Since the slope \(\mathrm{m} = -35\) is negative, the function is decreasing

Answer: B) Decreasing linear

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that "constant amount removed each day" creates a linear relationship rather than exponential.

Some students think that because water is being "removed" or "reduced," this automatically means exponential decay. They associate any decrease with exponential functions from contexts like radioactive decay or compound interest. However, exponential functions involve constant percentage changes, not constant amount changes.

This may lead them to select Choice A (Decreasing exponential).

Second Most Common Error:

Poor TRANSLATE reasoning: Students correctly identify the function as linear but get confused about the direction.

They might think: "Water is being added each day" (misreading "removed" as "added") or reason that since time increases, the function must be increasing. They don't properly connect the negative coefficient (-35) with the decreasing nature of the function.

This may lead them to select Choice D (Increasing linear).

The Bottom Line:

This problem tests whether students can distinguish between constant rate changes (linear) versus constant percentage changes (exponential), and whether they can properly interpret the direction of change from the mathematical relationship.