Two nearby trees are perpendicular to the ground, which is flat. One of these trees is 10 feet tall and...

1. TRANSLATE the problem information

• Given information:
  • Tree 1: 10 feet tall, 5-foot shadow
  • Tree 2: unknown height, 2-foot shadow
  • Both measurements taken at the same time

2. INFER the mathematical relationship

• Since both trees are measured at the same time of day, the sun hits them at the same angle
• This creates similar right triangles (tree height and shadow length as legs)
• For similar triangles, corresponding sides are proportional

3. TRANSLATE this insight into a proportion

• Set up the ratio: \(\mathrm{height_1/shadow_1 = height_2/shadow_2}\)
• Substitute known values: \(\mathrm{10/5 = x/2}\)

4. SIMPLIFY the proportion to solve for x

• First simplify the left side: \(\mathrm{10/5 = 2}\)
• So we have: \(\mathrm{2 = x/2}\)
• Multiply both sides by 2: \(\mathrm{x = 4}\)

Answer: B. 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize this as a similar triangles situation. Instead, they might try simple arithmetic operations like subtraction or addition with the given numbers, leading to answers like \(\mathrm{10 - 5 - 2 = 3}\). This may lead them to select Choice A (3).

Second Most Common Error:

Poor TRANSLATE reasoning: Students set up incorrect relationships, such as comparing differences rather than ratios. They might calculate \(\mathrm{10 - 2 = 8}\) (subtracting the shadow lengths from the tree height). This may lead them to select Choice C (8).

The Bottom Line:

This problem tests whether students can recognize proportional relationships in real-world contexts. The key insight is seeing that shadows and heights maintain constant ratios when measured simultaneously.