Let \(\mathrm{B(h) = 60(0.9)^{(h/6)} + 20}\). The function B gives the predicted remaining battery capacity \(\mathrm{B(h)}\), in percent, of a...

1. TRANSLATE the mathematical statement

• Given statement: "B(18) is approximately equal to 64"
• In function notation: \(\mathrm{B(18) ≈ 64}\)
• What this tells us:
  • The input to function B is 18
  • The output from function B is approximately 64

2. TRANSLATE the context of the function

• From \(\mathrm{B(h) = 60(0.9)^{(h/6)} + 20}\):
  • Input \(\mathrm{h}\) = months after smartphone is purchased
  • Output \(\mathrm{B(h)}\) = predicted remaining battery capacity in percent

3. INFER the complete interpretation

• Combining the mathematical statement with context:
  • Input: \(\mathrm{h = 18}\) means "18 months after purchase"
  • Output: \(\mathrm{B(18) ≈ 64}\) means "approximately 64% battery capacity"
• Therefore: The battery capacity is approximately 64% after 18 months

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse input and output in function notation, thinking \(\mathrm{B(18) = 64}\) means "18% battery capacity at 64 months after purchase."

This fundamental misunderstanding of function notation leads them to swap the numerical values and their meanings. They see the numbers 18 and 64 but don't properly identify which is the input (goes inside parentheses) and which is the output (the function value).

This may lead them to select Choice A (18% at 64 months).

Second Most Common Error:

Poor TRANSLATE reasoning: Students incorrectly think that since \(\mathrm{h/6}\) appears in the exponent of the function, the time input should be \(\mathrm{h/6 = 18/6 = 3}\) months instead of \(\mathrm{h = 18}\) months.

They misinterpret the mathematical structure, not realizing that \(\mathrm{h/6}\) is just part of the exponential expression, while \(\mathrm{h}\) itself remains the input variable representing months after purchase.

This may lead them to select Choice C (64% at 18/6 months).

The Bottom Line:

This problem tests whether students truly understand function notation in context. The key insight is that in \(\mathrm{f(input) = output}\), the value inside the parentheses is always the input, and the resulting function value is always the output—regardless of how complex the function expression might be.