A right circular cone with its vertex pointing downwards has a height of 30 centimeters. The cone is filled with...

1. TRANSLATE the problem information

• Given information:
  • Right circular cone with vertex pointing downwards
  • Total height of cone: \(\mathrm{H = 30\text{ cm}}\)
  • Height of water from vertex: \(\mathrm{h = 20\text{ cm}}\)
  • Radius of water surface: \(\mathrm{r = 14\text{ cm}}\)
  • Find: radius of cone's base (R)

2. INFER the geometric relationship

• Key insight: Since the vertex points downward, the water forms a smaller cone inside the larger cone
• Both the water cone and the full cone share the same vertex and have the same shape
• This means they are similar figures with proportional corresponding dimensions
• The ratio of heights equals the ratio of radii: \(\mathrm{\frac{h}{H} = \frac{r}{R}}\)

3. TRANSLATE this relationship into an equation

• Set up the proportion: \(\mathrm{\frac{20}{30} = \frac{14}{R}}\)

4. SIMPLIFY to solve for R

• Reduce the fraction: \(\mathrm{\frac{2}{3} = \frac{14}{R}}\)
• Cross-multiply: \(\mathrm{2 \times R = 3 \times 14}\)
• Calculate: \(\mathrm{2R = 42}\)
• Divide both sides by 2: \(\mathrm{R = 21}\)

Answer: C) 21

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not recognize that the water and cone form similar shapes, instead trying to use volume formulas or other complex relationships.

Without this key insight, they might attempt to calculate volumes or try to work with the cone's slant height, leading to unnecessary complexity and likely errors. This often leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor TRANSLATE reasoning: Students might misinterpret which measurements correspond to which parts of the similar triangles, perhaps confusing the water height with the remaining empty space \(\mathrm{(30 - 20 = 10\text{ cm})}\).

Setting up an incorrect proportion like \(\mathrm{\frac{10}{30} = \frac{14}{R}}\) would give \(\mathrm{R = 42}\), leading them to select Choice E (42).

The Bottom Line:

This problem tests whether students can recognize similarity relationships in three-dimensional figures. The key breakthrough is seeing that a partially filled cone creates two similar cones sharing the same vertex.