The function h is defined by \(\mathrm{h(w) = 5w^3 - 15}\). What is the value of w when \(\mathrm{h(w) =...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{h(w) = 5w^3 - 15}\)
  • Condition: \(\mathrm{h(w) = 120}\)
  • Need to find: value of w

• This means we need to solve: \(\mathrm{5w^3 - 15 = 120}\)

2. SIMPLIFY to isolate the variable term

• Add 15 to both sides:
  \(\mathrm{5w^3 - 15 + 15 = 120 + 15}\)
  \(\mathrm{5w^3 = 135}\)

• Divide both sides by 5:
  \(\mathrm{w^3 = 27}\)

3. SIMPLIFY to find the final answer

• Take the cube root of both sides:
  \(\mathrm{w = \sqrt[3]{27} = 3}\)

• Verify: \(\mathrm{h(3) = 5(3)^3 - 15 = 5(27) - 15 = 135 - 15 = 120}\) ✓

Answer: B (3)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may confuse the setup, trying to solve \(\mathrm{h(w) = 5w^3 - 15 = 120}\) by setting the function equal to both expressions simultaneously, or they might substitute incorrectly.

This leads to confusion and abandoning systematic solution, resulting in guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{5w^3 - 15 = 120}\) but make arithmetic errors, such as:

• Getting \(\mathrm{w^3 = 35}\) instead of \(\mathrm{w^3 = 27}\) (from incorrect division: \(\mathrm{135 \div 5 = 35}\))
• Not recognizing that \(\mathrm{\sqrt[3]{27} = 3}\), leading them to estimate incorrectly

Arithmetic errors may lead them to select Choice A (2) or get confused and guess.

The Bottom Line:

This problem tests whether students can translate function notation into equations and perform accurate algebraic manipulations, particularly with cube roots. Success depends on methodical equation solving rather than trying to work backwards from answer choices.