The function r is defined by \(\mathrm{r(x) = \frac{56}{x^3}}\). For which value of x is \(\mathrm{r(x) = 7}\)?

1. TRANSLATE the problem information

• Given information:
  • Function \(\mathrm{r(x) = \frac{56}{x^3}}\)
  • Need to find x where \(\mathrm{r(x) = 7}\)
• What this tells us: We need to set up an equation where the function output equals 7

2. INFER the solution approach

• Since we want \(\mathrm{r(x) = 7}\), we substitute the function definition
• This gives us: \(\mathrm{\frac{56}{x^3} = 7}\)
• Strategy: Solve this equation algebraically for x

3. SIMPLIFY to solve the equation

• Start with: \(\mathrm{\frac{56}{x^3} = 7}\)
• Multiply both sides by \(\mathrm{x^3}\): \(\mathrm{56 = 7x^3}\)
• Divide both sides by 7: \(\mathrm{8 = x^3}\)
• Take the cube root: \(\mathrm{x = \sqrt[3]{8} = 2}\)

4. Verify the answer

• Check: \(\mathrm{r(2) = \frac{56}{2^3} = \frac{56}{8} = 7}\) ✓

Answer: A. 2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when working with cube roots or fractions.

Common mistakes include thinking \(\mathrm{\sqrt[3]{8} = 4}\) (confusing with square root) or making calculation errors with \(\mathrm{\frac{56}{8}}\). Some students might also incorrectly manipulate the fraction, perhaps trying to "flip" it without proper justification.

This may lead them to select Choice B (8/3) or Choice D (8) depending on their specific calculation error.

Second Most Common Error:

Inadequate TRANSLATE reasoning: Students might misinterpret what the problem is asking for.

Some students might plug in answer choices without setting up the equation systematically, or they might confuse the function input and output, thinking they need to find \(\mathrm{r(x)}\) when \(\mathrm{x = 7}\) instead of finding x when \(\mathrm{r(x) = 7}\).

This leads to confusion and random answer selection among the choices.

The Bottom Line:

This problem requires careful algebraic manipulation with fractions and cube roots. The most critical skill is recognizing that function problems often reduce to equation-solving problems, then executing the algebra accurately.