A rectangle has width y and length 3y + k, where k is a constant. The area of the rectangle...

1. TRANSLATE the problem information

• Given information:
  • Rectangle width: \(\mathrm{y}\)
  • Rectangle length: \(\mathrm{3y + k}\)
  • Rectangle area: \(\mathrm{3y^2 + 11y}\) square units
• What we need: Find the value of \(\mathrm{k}\)

2. TRANSLATE the area relationship

• Area of rectangle = width × length
• So: \(\mathrm{3y^2 + 11y = y \times (3y + k)}\)

3. SIMPLIFY the right side of the equation

• Expand using distributive property: \(\mathrm{y(3y + k) = 3y^2 + ky}\)
• Our equation becomes: \(\mathrm{3y^2 + 11y = 3y^2 + ky}\)

4. INFER the solution strategy

• Key insight: Since this equation must be true for ALL real numbers \(\mathrm{y}\), the polynomials on both sides must be identical
• This means corresponding coefficients must be equal

5. APPLY CONSTRAINTS to match coefficients

• Coefficient of \(\mathrm{y^2}\) terms: \(3 = 3\) ✓ (This checks out)
• Coefficient of \(\mathrm{y}\) terms: \(11 = \mathrm{k}\)
• Therefore: \(\mathrm{k = 11}\)

Answer: D. 11

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may incorrectly set up the area equation, perhaps writing it as \(\mathrm{(3y + k) \times y = 3y^2 + 11y + k}\) or confusing which expression represents length vs. width.

This fundamental setup error makes the entire solution impossible and leads to confusion and guessing.

Second Most Common Error:

Missing conceptual knowledge about polynomial equality: Students might correctly expand to get \(\mathrm{3y^2 + ky = 3y^2 + 11y}\) but fail to recognize that coefficients of like terms must be equal. Instead, they might try to solve for specific values of \(\mathrm{y}\) or attempt to isolate \(\mathrm{k}\) incorrectly.

This may lead them to select Choice A (3) by incorrectly thinking \(\mathrm{k}\) comes from the coefficient of \(\mathrm{y^2}\) in the length expression.

The Bottom Line:

This problem tests whether students can translate a geometric relationship into algebra and then apply the fundamental principle that equal polynomials have equal coefficients. The key breakthrough is recognizing that "for all real numbers \(\mathrm{y}\)" means the polynomials must be identical term by term.