Question:A construction company rents a large crane. The total rental cost consists of a fixed one-time fee plus a constant...

1. TRANSLATE the problem information

• Given information:
  • Fixed one-time fee + constant daily charge = total cost
  • 3 days costs $210
  • 7 days costs $430
  • Need function C(d) for d days

• What this tells us: We have two coordinate points (3, 210) and (7, 430) from a linear function

2. INFER the mathematical approach

• Since we have a fixed fee plus daily charges, this creates a linear relationship
• Linear functions have the form \(\mathrm{C(d) = md + b}\) where:
  • \(\mathrm{m = slope}\) = daily rental charge
  • \(\mathrm{b = y\text{-}intercept}\) = fixed one-time fee
• Strategy: Find slope first using our two points, then find the y-intercept

3. SIMPLIFY to find the daily charge (slope)

Using the slope formula with points (3, 210) and (7, 430):

• \(\mathrm{m = \frac{430 - 210}{7 - 3} = \frac{220}{4} = 55}\)
• The daily rental charge is $55

4. SIMPLIFY to find the fixed fee (y-intercept)

Substitute the slope and one point into \(\mathrm{C(d) = 55d + b}\):

• Using point (3, 210): \(\mathrm{210 = 55(3) + b}\)
• \(\mathrm{210 = 165 + b}\)
• \(\mathrm{b = 45}\)
• The fixed one-time fee is $45

5. Write the complete function

• \(\mathrm{C(d) = 55d + 45}\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may struggle to extract the two coordinate points from the word problem or fail to recognize that 'fixed fee plus daily charge' creates the linear form \(\mathrm{y = mx + b}\).

Without clear coordinate points, they might guess among answer choices or try to work backwards from the choices without a systematic approach. This leads to confusion and guessing rather than methodical solution.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the approach but make arithmetic errors when calculating 220 ÷ 4 = 55 for the slope, or when solving 210 = 165 + b for the y-intercept.

These calculation errors can lead them to select Choice A (\(\mathrm{C(d) = 50d + 60}\)) or Choice C (\(\mathrm{C(d) = 60d + 30}\)) depending on which arithmetic step went wrong.

The Bottom Line:

This problem tests whether students can bridge the gap between a real-world linear situation and its mathematical representation. The key insight is recognizing that 'fixed fee + daily charge' automatically means linear function, and that the given costs provide coordinate points for finding that function.