A square has side length s given by s = sqrt(8t). If s = 12, what is the value of...

1. TRANSLATE the problem information

• Given information:
  • Side length formula: \(\mathrm{s = \sqrt{8t}}\)
  • Current side length: \(\mathrm{s = 12}\)
  • Find: \(\mathrm{9t}\)

2. TRANSLATE to create an equation

• Since \(\mathrm{s = 12}\), substitute this value into the formula:
  \(\mathrm{\sqrt{8t} = 12}\)
• Now we have a square root equation to solve

3. SIMPLIFY by eliminating the square root

• Square both sides of the equation:
  \(\mathrm{(\sqrt{8t})^2 = 12^2}\)
  \(\mathrm{8t = 144}\)

4. SIMPLIFY to solve for t

• Divide both sides by 8:
  \(\mathrm{t = 144 ÷ 8 = 18}\)

5. TRANSLATE to find what the question asks for

• The question asks for \(\mathrm{9t}\), not just \(\mathrm{t}\)
• Calculate: \(\mathrm{9t = 9 × 18 = 162}\)

Answer: C (162)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students solve correctly for \(\mathrm{t = 18}\) but forget that the question asks for \(\mathrm{9t}\), not \(\mathrm{t}\).

They complete all the algebra correctly, find \(\mathrm{t = 18}\), and then select an answer choice with 18 in it. Since 18 isn't among the choices, they might double-check their arithmetic but still miss that they need to multiply by 9.

This leads to confusion and guessing among the available choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when squaring 12 or dividing 144 by 8.

For example, they might calculate \(\mathrm{12^2 = 124}\) instead of 144, or divide incorrectly. These calculation errors cascade through the remaining steps, leading them to select incorrect answer choices or become confused when their final answer doesn't match any option.

The Bottom Line:

This problem tests whether students can systematically work through a multi-step algebraic process while staying focused on what the question actually asks for. The key challenge is remembering that finding \(\mathrm{t}\) is just the intermediate step—the real goal is finding \(\mathrm{9t}\).