A community program spends money from a grant at a constant daily rate. The equation B = 2400 - 150t...

1. INFER what the t-intercept represents

• The t-intercept occurs when \(\mathrm{B = 0}\)
• In this context, \(\mathrm{B = 0}\) means no money remains in the grant
• So the t-intercept tells us when the grant money runs out completely

2. SIMPLIFY to find the t-intercept

• Set \(\mathrm{B = 0}\) in the equation \(\mathrm{B = 2400 - 150t}\):

\(\mathrm{0 = 2400 - 150t}\)

• Add \(\mathrm{150t}\) to both sides:

\(\mathrm{150t = 2400}\)

• Divide both sides by 150:

\(\mathrm{t = \frac{2400}{150} = 16}\)

3. TRANSLATE the result back to context

• \(\mathrm{t = 16}\) means after 16 days, the balance \(\mathrm{B = 0}\)
• This means the program will have spent the entire grant in about 16 days

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect that "t-intercept" means "when \(\mathrm{B = 0}\)" and what this represents contextually. They may know how to find where the line crosses the t-axis mathematically, but fail to interpret that \(\mathrm{B = 0}\) means the money is gone.

This leads to confusion about what the value \(\mathrm{t = 16}\) actually means, causing them to guess or select a distractor that sounds reasonable but doesn't match the intercept interpretation.

Second Most Common Error:

Poor TRANSLATE reasoning: Students correctly find \(\mathrm{t = 16}\) but then misinterpret this value. They might see "16" and think it relates to the daily spending rate rather than the time when funds are exhausted.

This may lead them to select Choice (C) (spending $16 per day) because they confuse the t-intercept value with the spending rate.

The Bottom Line:

This problem tests whether students can connect the abstract concept of an intercept to its real-world meaning. Finding \(\mathrm{t = 16}\) is straightforward algebra, but understanding that this represents "when the money runs out" requires conceptual insight about what \(\mathrm{B = 0}\) means in context.