A rectangular garden plot has a total area of 840 square feet available for planting. The gardener plans to reserve...

1. TRANSLATE the problem information

• Given information:
  • Total garden area: 840 square feet
  • Tool shed reservation: 35 square feet
  • Tomato plant space requirement: 12 square feet each
  • Pepper plant space requirement: 18 square feet each
  • Variables: \(\mathrm{t}\) = number of tomato plants, \(\mathrm{p}\) = number of pepper plants

2. INFER the key constraint relationship

• The space used by all plants cannot exceed the available planting space
• This creates an inequality constraint, not an equation
• We need to find available space first, then set up the constraint

3. Calculate available planting area

Available area = Total area - Reserved area
Available area = \(\mathrm{840 - 35 = 805}\) square feet

4. TRANSLATE plant space usage into mathematical expression

• Space used by \(\mathrm{t}\) tomato plants: \(\mathrm{12t}\) square feet
• Space used by \(\mathrm{p}\) pepper plants: \(\mathrm{18p}\) square feet
• Total space used by plants: \(\mathrm{12t + 18p}\) square feet

5. APPLY CONSTRAINTS to create the inequality

• Total plant space \(\mathrm{\leq}\) Available space
• \(\mathrm{12t + 18p \leq 805}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Process Skill Gap - Weak TRANSLATE reasoning: Students correctly identify the space requirements but fail to subtract the tool shed area from total area, using 840 instead of 805.

They set up: \(\mathrm{12t + 18p \leq 840}\) or similar variations.

This reasoning error leads them to select Choice C (\(\mathrm{18t + 12p \leq 840}\)) if they also mix up the coefficients, or creates confusion if this exact form isn't available.

Second Most Common Error:

Process Skill Gap - Poor APPLY CONSTRAINTS execution: Students correctly calculate available space (805) and set up the expression (\(\mathrm{12t + 18p}\)) but use "greater than or equal to" instead of "less than or equal to."

They reason: "We want to use at least this much space" rather than "We cannot use more than this much space."

This leads them to select Choice B (\(\mathrm{12t + 18p \geq 805}\)).

The Bottom Line:

This problem tests whether students can distinguish between total capacity and available capacity, then properly apply constraint logic. The key insight is that real-world space limitations create upper bounds (\(\mathrm{\leq}\)), not lower bounds (\(\mathrm{\geq}\)).