A vertical pole stands on level ground. A straight support beam leans from the ground to a point 24 feet...

1. TRANSLATE the problem information

• Given information:
  • Vertical pole on level ground
  • Support beam from ground to point 24 feet up the pole
  • Beam makes \(38°\) angle with ground
  • Forms a right triangle with pole and ground

• What this tells us: We have a right triangle where the vertical pole and ground meet at \(90°\), with the beam as the hypotenuse.

2. INFER the trigonometric setup

• The key insight: We need to identify which sides are opposite and adjacent to the \(38°\) angle
• The \(38°\) angle is between the beam (hypotenuse) and the ground
• Therefore: the vertical height (24 feet) is opposite to \(38°\), and the horizontal ground distance is adjacent to \(38°\)
• We need the base length to find area, so we'll use: \(\tan(38°) = \frac{\mathrm{opposite}}{\mathrm{adjacent}}\)

3. SIMPLIFY to find the base length

• From \(\tan(38°) = \frac{24}{\mathrm{base}}\), we get:
  \(\mathrm{base} = \frac{24}{\tan(38°)}\)
  \(\mathrm{base} = 24 \times \frac{1}{\tan(38°)}\)
  \(\mathrm{base} = 24\cot(38°)\)

4. SIMPLIFY to find the area

• Area = (1/2) × base × height
• \(\mathrm{Area} = \frac{1}{2} \times (24\cot 38°) \times 24\)
• \(\mathrm{Area} = \frac{1}{2} \times 576 \times \cot 38°\)
• \(\mathrm{Area} = 288\cot 38°\)

• Since the area is in the form \(k\cot 38°\), we have \(k = 288\)

Answer: 288

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students incorrectly identify which side is opposite vs adjacent to the \(38°\) angle, thinking the base (ground distance) is opposite to \(38°\) instead of the height.

This leads them to set up: \(\tan(38°) = \frac{\mathrm{base}}{24}\), giving \(\mathrm{base} = 24\tan(38°)\) instead of \(\mathrm{base} = 24\cot(38°)\). When they calculate the area, they get \(\mathrm{Area} = \frac{1}{2} \times (24\tan 38°) \times 24 = 288\tan 38°\). Since they need the form \(k\cot 38°\), they might think \(k = \frac{288}{[\cot(38°)/\tan(38°)]} = \frac{288}{1}\) and get confused, or they might try to convert tan to cot incorrectly.

This leads to confusion and abandoning systematic solution for guessing.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly set up \(\tan(38°) = \frac{24}{\mathrm{base}}\) and find \(\mathrm{base} = 24\cot(38°)\), but make arithmetic errors when computing the area.

They might calculate: \(\mathrm{Area} = \frac{1}{2} \times (24\cot 38°) \times 24 = 24 \times 24 \times \cot 38° = 576\cot 38°\), forgetting the \(\frac{1}{2}\) factor. This gives them \(k = 576\) instead of \(k = 288\).

The Bottom Line:

This problem requires clear spatial reasoning to set up the right triangle correctly and identify which trigonometric relationship to use. The algebraic manipulation is straightforward once the setup is correct, but the geometric visualization is crucial for success.