x = 64y = sqrt[3]{x + 12}The graphs of the given equations intersect at the point \(\mathrm{(x, y)}\) in the...

1. TRANSLATE the intersection condition

• Given information:
  • \(\mathrm{x = 64}\)
  • \(\mathrm{y = \sqrt[3]{x} + 12}\)
  • These graphs intersect at point \(\mathrm{(x, y)}\)

• What this tells us: Since the point \(\mathrm{(x, y)}\) lies on both graphs, it must satisfy both equations simultaneously

2. TRANSLATE the solution strategy

• We know \(\mathrm{x = 64}\), so we can substitute this value into the y equation
• This will give us the y-coordinate of the intersection point

3. SIMPLIFY through substitution and calculation

• Substitute \(\mathrm{x = 64}\) into \(\mathrm{y = \sqrt[3]{x} + 12}\):
  \(\mathrm{y = \sqrt[3]{64} + 12}\)

• Evaluate the cube root:
  \(\mathrm{\sqrt[3]{64} = 4}\) (since \(\mathrm{4^3 = 4 \times 4 \times 4 = 64}\))

• Complete the addition:
  \(\mathrm{y = 4 + 12 = 16}\)

Answer: B (16)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak conceptual knowledge about cube roots: Students may confuse cube roots with square roots or not recognize that \(\mathrm{\sqrt[3]{64} = 4}\).

Some students might calculate \(\mathrm{\sqrt{64} = 8}\) instead of \(\mathrm{\sqrt[3]{64} = 4}\), leading to \(\mathrm{y = 8 + 12 = 20}\). This may lead them to select Choice C (20).

Second Most Common Error Path:

Poor TRANSLATE reasoning: Students might not understand what "intersection point" means and attempt to solve both equations as a system rather than recognizing they should substitute the given x-value.

This leads to confusion about the problem setup and may cause them to get stuck and guess randomly.

The Bottom Line:

This problem tests whether students understand intersection points and can work with cube roots. The key insight is recognizing that intersection means substitution, and then accurately evaluating \(\mathrm{\sqrt[3]{64} = 4}\).