\(\mathrm{B(t) = -15{,}000t + 300{,}000}\)The given function B models the balance, in dollars, of a startup company's initial funding t...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{B(t) = -15,000t + 300,000}\)
  • \(\mathrm{B(t)}\) represents balance in dollars
  • \(\mathrm{t}\) represents months after starting operations
  • Question asks: amount spent each month
• What this tells us: We need to find the rate at which money leaves the account each month.

2. INFER the mathematical approach

• This is a linear function in the form \(\mathrm{y = mx + b}\)
• In linear functions, the slope (coefficient of the variable) represents the rate of change
• Since we want the monthly spending rate, we need to examine the slope

3. INFER the meaning of each component

• \(\mathrm{Slope = -15,000}\) (coefficient of t)
• \(\mathrm{Y\text{-}intercept = 300,000}\) (constant term)
• The slope tells us how much the balance changes for each unit increase in time (each month)

4. INFER the real-world interpretation

• A slope of -15,000 means the balance decreases by $15,000 every month
• When a company's balance decreases, that money is being spent
• Therefore, the company spends $15,000 each month

Answer: B (15,000)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students correctly identify the function structure but misinterpret what the slope represents in context. They might think the negative slope means the company is losing money in a bad way, or they might focus on the y-intercept (300,000) thinking that's the monthly amount because it's the largest number.

This may lead them to select Choice D (300,000) by focusing on the initial funding amount rather than the rate of change.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misread the question and think they need to find how much money the company has initially or at some specific time, rather than the monthly spending rate. They might also confuse "spending" with "earning" and interpret the negative slope incorrectly.

This may lead them to select Choice A (1,500) through incorrect calculations or Choice C (285,000) by performing unnecessary arithmetic with the given values.

The Bottom Line:

The key insight is recognizing that in linear function word problems, the slope always represents the rate of change, and in financial contexts, you must carefully interpret whether that change represents spending, earning, or other financial flows. The negative sign is crucial—it indicates money flowing out (spending) rather than in (earning).