A rectangular aquarium has a base measuring 24,cm by 50,cm. The aquarium contains water to a height of 15,cm. When...

1. TRANSLATE the problem information

• Given information:
  • Aquarium base: \(24 \text{ cm} \times 50 \text{ cm}\) rectangle
  • Initial water height: \(15 \text{ cm}\)
  • Final water height after rock added: \(20 \text{ cm}\)
  • Need to find: volume of the rock

2. INFER the key relationship

• When the rock is submerged, it displaces water
• The volume of displaced water equals the volume of the rock
• The displaced water forms a rectangular prism with the same base as the aquarium
• The height of this displaced water prism equals the rise in water level

3. SIMPLIFY to find the base area

• Base area = length × width = \(24 \text{ cm} \times 50 \text{ cm} = 1{,}200 \text{ cm}^2\)

4. SIMPLIFY to find the height of displaced water

• Height increase = Final height - Initial height = \(20 \text{ cm} - 15 \text{ cm} = 5 \text{ cm}\)

5. SIMPLIFY to find the rock's volume

• Volume of displaced water = Base area × Height increase
• Volume = \(1{,}200 \text{ cm}^2 \times 5 \text{ cm} = 6{,}000 \text{ cm}^3\)

Answer: B. 6,000

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the displacement relationship and instead try to calculate the total volume of water or the aquarium volume.

They might calculate the total water volume (\(24 \times 50 \times 20 = 24{,}000 \text{ cm}^3\)) and think this is the rock volume, leading them to select Choice D (24,000). Or they calculate the initial water volume (\(24 \times 50 \times 15 = 18{,}000 \text{ cm}^3\)) and select Choice C (18,000).

Second Most Common Error:

Poor SIMPLIFY execution: Students understand the displacement concept but make calculation errors, particularly when finding the base area.

They might calculate \(24 \times 5 = 120\) instead of using the full base area calculation, leading to \(120 \times 5 = 600\), causing confusion and guessing among the provided choices.

The Bottom Line:

This problem tests whether students can connect a real-world displacement scenario to the mathematical concept that displaced volume equals object volume. The key insight is recognizing that only the change in water level matters, not the absolute water levels.