Two identical cubes, each with an edge length of 3 centimeters, are glued together to form a right rectangular prism....

1. VISUALIZE the problem setup

• Given information:
  • Two identical cubes, each with edge length 3 cm
  • Cubes are glued together face-to-face
  • Need to find surface area of resulting rectangular prism

• VISUALIZE what happens: When you place two cubes side by side and glue them together, you get a rectangular box that's longer than it is wide or tall.

2. INFER the solution approach

• Key insight: When faces are glued together, those faces are now inside the shape and don't count toward surface area
• Two solution paths are available:
  • Calculate dimensions of new prism and use rectangular prism formula
  • Start with total surface area of separate cubes and subtract covered faces

3. TRANSLATE dimensions of the new rectangular prism

Method 1 approach:

• Length: \(3 + 3 = 6\) cm (two cubes placed end-to-end)
• Width: 3 cm (unchanged from original cube)
• Height: 3 cm (unchanged from original cube)

4. SIMPLIFY using the rectangular prism formula

• Surface area formula: \(\mathrm{SA} = 2(\mathrm{lw} + \mathrm{lh} + \mathrm{wh})\)

\(\mathrm{SA} = 2(6 \times 3 + 6 \times 3 + 3 \times 3)\)

\(\mathrm{SA} = 2(18 + 18 + 9)\)

\(\mathrm{SA} = 2(45) = 90\) square centimeters

Answer: C. 90

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak VISUALIZE skill: Students don't properly visualize what happens when cubes are joined, thinking the surface area simply doubles or stays the same as one cube.

Some students calculate \(2 \times 54 = 108\) (just adding surface areas without accounting for covered faces), leading them to select Choice D (108).

Second Most Common Error:

Inadequate INFER reasoning: Students understand that some area is lost but incorrectly calculate how much. They might subtract only one face area (9 sq cm) instead of two, getting \(108 - 9 = 99\), which isn't an option, causing confusion and guessing.

The Bottom Line:

This problem tests spatial reasoning combined with surface area concepts. Success requires visualizing the 3D transformation and understanding that internal faces don't contribute to surface area.