A parking garage charges a fee based solely on time parked. The total cost \(\mathrm{C(t)}\), in dollars, is equal to...

1. TRANSLATE the problem information

• Given information:
  • Base parking fee: \(\$3\)
  • Additional cost: \(\$2.50\) per hour parked
  • Need to find: what type of function describes \(\mathrm{C(t)}\)

• What this tells us: We need to write a function that adds the base fee to the hourly charges

2. INFER the mathematical approach

• Since we have a fixed base amount plus a rate per hour, this suggests we're building a function of the form: \(\mathrm{base + (rate \times time)}\)
• We need to write this as \(\mathrm{C(t)}\) = something, then identify its characteristics

3. TRANSLATE into function notation

• Base fee: \(\$3\)
• Rate per hour: \(\$2.50\)
• Time: \(\mathrm{t}\) hours
• Total cost: \(\mathrm{C(t) = 3 + 2.50t = 2.5t + 3}\)

4. INFER the function type and behavior

• Our function \(\mathrm{C(t) = 2.5t + 3}\) has the form \(\mathrm{mt + b}\) where:
  • \(\mathrm{m = 2.5}\) (coefficient of \(\mathrm{t}\))
  • \(\mathrm{b = 3}\) (constant term)
• This is the standard form of a linear function
• Since \(\mathrm{m = 2.5 \gt 0}\), the function increases as \(\mathrm{t}\) increases
• This is NOT exponential because \(\mathrm{t}\) is not an exponent

Answer: D (Increasing linear)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students struggle to convert "3 plus 2.50 per hour" into proper mathematical notation, potentially writing something like \(\mathrm{C(t) = 3 \times 2.50t}\) or getting confused about how to combine the base fee with the hourly rate.

This leads to an incorrect function form, making it impossible to correctly identify the function type and behavior.

Second Most Common Error:

Conceptual confusion about function types: Students may think that any function involving time must be exponential, or they might confuse the defining characteristics of linear vs exponential functions.

They might see the variable \(\mathrm{t}\) and assume it must be exponential, leading them to select Choice C (Increasing exponential) without properly analyzing the function form.

The Bottom Line:

This problem tests whether students can translate a real-world rate structure into mathematical notation and then classify the resulting function. Success depends on understanding that linear functions have constant rates of change (like \(\$2.50\) per hour), while exponential functions involve repeated multiplication by a base.