The buildings of a shopping center are designed to allow water to drain from the roof into gutters on the...

1. TRANSLATE the problem information

• Given information:
  • Table shows relationship between roof area x (square feet) and water drainage f(x) (gallons)
  • Data points: \(\mathrm{(2,520, 4,536)}\), \(\mathrm{(3,780, 6,804)}\), \(\mathrm{(5,040, 9,072)}\)
  • All answer choices are in the form \(\mathrm{f(x) = mx}\)
• What this tells us: We need to find the constant rate m

2. INFER the approach

• Since all equations have the form \(\mathrm{f(x) = mx}\), the rate m can be found by calculating \(\mathrm{f(x)/x}\)
• If the relationship is truly linear, this ratio should be the same for all data points
• We'll calculate the rate using each data point to verify consistency

3. SIMPLIFY to find the constant rate

• Using first data point \(\mathrm{(2,520, 4,536)}\):
  Rate = \(\mathrm{4,536 \div 2,520 = 1.8}\) (use calculator)
• Using second data point \(\mathrm{(3,780, 6,804)}\):
  Rate = \(\mathrm{6,804 \div 3,780 = 1.8}\) (use calculator)
• Using third data point \(\mathrm{(5,040, 9,072)}\):
  Rate = \(\mathrm{9,072 \div 5,040 = 1.8}\) (use calculator)

4. INFER the final equation

• Since all ratios equal 1.8, the constant rate of change is 1.8
• Therefore: \(\mathrm{f(x) = 1.8x}\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that they need to find the constant rate of change by dividing f(x) by x. Instead, they might try to substitute one data point into each equation or look for patterns in the numbers themselves without understanding the underlying linear relationship.

This leads to confusion and random guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students understand they need to find the rate but make calculation errors when dividing the large numbers, or they only check one data point instead of verifying the rate is consistent across all points.

This may lead them to select Choice A (\(\mathrm{0.6x}\)) or other incorrect options based on miscalculated rates.

The Bottom Line:

This problem tests whether students can work backwards from data to identify the equation of a linear function. Success requires both recognizing the strategy (find the constant rate) and executing accurate calculations to verify that rate across multiple data points.