Question:The area of an equilateral triangle is 3600sqrt(3) square centimeters. The height of the triangle can be written in the...

1. TRANSLATE the problem information

• Given information:
  • Area of equilateral triangle = \(\mathrm{3600\sqrt{3}}\) square centimeters
  • Height can be written as \(\mathrm{k\sqrt{3}}\) centimeters
• Need to find: The value of k

2. INFER the solution strategy

• To find height, we need the side length first
• We can use the area formula to find the side length
• Then use the height formula to express height in terms of \(\mathrm{k\sqrt{3}}\)

3. TRANSLATE the area relationship into an equation

• Area formula for equilateral triangle: \(\mathrm{A = \frac{s^2\sqrt{3}}{4}}\)
• Set up equation: \(\mathrm{\frac{s^2\sqrt{3}}{4} = 3600\sqrt{3}}\)

4. SIMPLIFY to find the side length

• Divide both sides by \(\mathrm{\sqrt{3}}\): \(\mathrm{\frac{s^2}{4} = 3600}\)
• Multiply both sides by 4: \(\mathrm{s^2 = 14400}\)
• Take the square root: \(\mathrm{s = 120}\) cm (positive value since it's a length)

5. INFER and apply the height formula

• Height formula for equilateral triangle: \(\mathrm{h = \frac{s\sqrt{3}}{2}}\)
• Substitute \(\mathrm{s = 120}\): \(\mathrm{h = \frac{120\sqrt{3}}{2} = 60\sqrt{3}}\)

6. TRANSLATE to find k

• We found \(\mathrm{h = 60\sqrt{3}}\)
• Since height is given as \(\mathrm{k\sqrt{3}}\), we have \(\mathrm{k = 60}\)

Answer: 60

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that they need to find the side length first to determine the height.

Some students try to work directly with height formulas or get confused about the relationship between area and height. They might attempt to manipulate the given information without a clear strategy, leading to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic errors when solving \(\mathrm{\frac{s^2}{4} = 3600}\).

Common mistakes include forgetting to multiply by 4 (getting \(\mathrm{s^2 = 900}\) instead of \(\mathrm{14400}\)), taking the square root incorrectly (getting \(\mathrm{s = 30}\) instead of \(\mathrm{120}\)), or making arithmetic errors in the final height calculation. This leads to incorrect values of k like 15 or 30.

The Bottom Line:

This problem requires recognizing the connection between area and height through the side length - students who try shortcuts or don't systematically work through the relationships between these triangle properties often get stuck or make computational errors.