A cylindrical can containing pieces of fruit is filled to the top with syrup before being sealed. The base of...

1. TRANSLATE the problem information

• Given information:
  • Cylindrical can with base area \(= 75 \text{ cm}^2\)
  • Height of can \(= 10 \text{ cm}\)
  • Syrup volume needed to fill can \(= 110 \text{ cm}^3\)
  • Need to find: Volume of fruit pieces

2. INFER the volume relationship

• Key insight: The fruit and syrup together completely fill the can
• This means: \(\text{Total can volume} = \text{Fruit volume} + \text{Syrup volume}\)
• Therefore: \(\text{Fruit volume} = \text{Total can volume} - \text{Syrup volume}\)

3. Calculate the total volume of the can

• \(\text{Volume of cylinder} = \text{Base area} \times \text{height}\)
• \(\text{Total volume} = 75 \text{ cm}^2 \times 10 \text{ cm} = 750 \text{ cm}^3\)

4. SIMPLIFY to find the fruit volume

• \(\text{Fruit volume} = \text{Total volume} - \text{Syrup volume}\)
• \(\text{Fruit volume} = 750 \text{ cm}^3 - 110 \text{ cm}^3 = 640 \text{ cm}^3\)

Answer: C. 640 cm³

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the volume relationship between the container, fruit, and syrup. They may try to use the syrup volume directly in some formula or get confused about what the \(110 \text{ cm}^3\) represents.

Without understanding that the syrup fills the empty space around the fruit, students might subtract the fruit volume from the syrup volume, or try to find the fruit volume using only the base area and syrup volume. This leads to confusion and guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Students correctly find the total volume as \(750 \text{ cm}^3\) but then select this as their final answer, not realizing they need to subtract the syrup volume.

This misconception treats the total can volume as the fruit volume, leading them to select Choice D (\(750 \text{ cm}^3\)).

The Bottom Line:

This problem tests whether students can work with volume relationships in real-world contexts. The key challenge is recognizing that when objects are placed in a container and fluid is added, the fluid volume tells us about the remaining empty space, not about the objects themselves.