The given function \(\mathrm{f(x) = 2x + 30}\) represents the perimeter, in centimeters (cm), of an isosceles triangle with two...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f(x) = 2x + 30}\) represents the perimeter in cm
  • The triangle is isosceles with two equal sides of length \(\mathrm{x}\) cm
  • We need to find the base length

• What this tells us: We have a function that gives us the total perimeter, and we know two of the three sides.

2. INFER the relationship between the function and triangle structure

• Since it's an isosceles triangle with two equal sides of length \(\mathrm{x}\), the perimeter must be:
  \(\mathrm{x + x + base = 2x + base}\)

• The key insight: The given function \(\mathrm{f(x) = 2x + 30}\) represents this same perimeter, so we can set up the equation:
  \(\mathrm{2x + base = 2x + 30}\)

3. SIMPLIFY to solve for the base

• Subtract \(\mathrm{2x}\) from both sides:
  \(\mathrm{2x + base - 2x = 2x + 30 - 2x}\)
  \(\mathrm{base = 30}\)

• The base length is 30 cm.

Answer: C) 30

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students might misinterpret what the function represents or how to set up the perimeter equation for an isosceles triangle.

Some students think the function \(\mathrm{f(x) = 2x + 30}\) directly gives them the base length, leading them to believe the base is either \(\mathrm{2x}\) or just conclude it's related to the coefficient 2. This may lead them to select Choice A (2) by focusing on the coefficient, or they might get confused about what \(\mathrm{x}\) represents and guess.

Second Most Common Error:

Poor INFER reasoning: Students correctly set up that perimeter = \(\mathrm{2x + base}\) but fail to connect this with the given function.

They might think they need additional information to solve the problem, not realizing that setting \(\mathrm{2x + base = 2x + 30}\) immediately gives them the answer. This leads to confusion and guessing among the answer choices.

The Bottom Line:

This problem tests whether students can bridge the gap between a mathematical function and its geometric meaning. The key breakthrough is recognizing that the function directly encodes the triangle's structure, making the solution surprisingly straightforward once the connection is made.