An architect is designing a wheelchair ramp that will lead up to the entrance of a library. The ramp forms...

1. TRANSLATE the problem information

• Given information:
  • The ramp, ground, and wall form a right triangle
  • Angle α is between the ramp and ground
  • \(\cos \alpha = \frac{3}{5}\)
  • Horizontal distance from start of ramp to wall = 60 feet
  • Need to find: length of ramp (hypotenuse)

2. INFER the approach

• The key insight: we can use the cosine ratio directly since we know the angle's cosine value and the adjacent side
• We don't need to find the opposite side or the actual angle measure
• Set up: \(\cos \alpha = \frac{\mathrm{adjacent}}{\mathrm{hypotenuse}}\)

3. TRANSLATE into the equation

• \(\cos \alpha = \frac{\mathrm{adjacent}}{\mathrm{hypotenuse}}\) becomes:
• \(\frac{3}{5} = \frac{60}{\mathrm{hypotenuse}}\)

4. SIMPLIFY to solve for the hypotenuse

• Cross multiply: \(3 \times \mathrm{hypotenuse} = 5 \times 60\)
• \(3 \times \mathrm{hypotenuse} = 300\)
• \(\mathrm{hypotenuse} = 300 \div 3 = 100\)

Answer: C. 100

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students often confuse which side is which in the right triangle setup. They might think the 60 feet is the hypotenuse or the opposite side instead of recognizing it as the adjacent side to angle α.

When they set up \(\cos \alpha = \frac{60}{\mathrm{hypotenuse}}\) (thinking 60 is the hypotenuse), they get \(\frac{3}{5} = \frac{60}{\mathrm{hypotenuse}}\), leading to \(\mathrm{hypotenuse} = 60 \times \frac{3}{5} = 36\).

This leads them to select Choice A (36).

Second Most Common Error:

Poor INFER strategy: Students recognize the setup correctly but think they need to find the actual angle first, then use it somehow. They might try to calculate \(\alpha = \arccos(\frac{3}{5}) \approx 53.13°\), then get confused about what to do next.

This causes them to get stuck and abandon their systematic approach, often guessing among the remaining choices.

The Bottom Line:

The challenge is recognizing that trigonometric ratios can be used directly with given ratio values - you don't always need to find the actual angle measure. The problem gives you everything needed to apply the cosine definition immediately.