A car-sharing service charges a base fee of $3 plus a daily surcharge that doubles each day, starting at $4...

1. TRANSLATE the problem request

• Given information:
  • Function model: \(\mathrm{C(d) = 3 + 4 \cdot 2^d}\)
  • Need to find the value k when \(\mathrm{d = 3}\)
• What this tells us: We need to evaluate \(\mathrm{C(3)}\) using the given formula

2. SIMPLIFY by substituting and calculating

• Substitute \(\mathrm{d = 3}\) into \(\mathrm{C(d) = 3 + 4 \cdot 2^d}\):
  \(\mathrm{C(3) = 3 + 4 \cdot 2^3}\)

• Calculate the exponent first: \(\mathrm{2^3 = 8}\)

• Continue the calculation:
  \(\mathrm{C(3) = 3 + 4 \cdot 8}\)
  \(\mathrm{C(3) = 3 + 32}\)
  \(\mathrm{C(3) = 35}\)

Answer: D (35)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Incorrectly calculating \(\mathrm{2^3}\) or making arithmetic errors in the multi-step calculation

Students might calculate \(\mathrm{2^3 = 6}\) (confusing exponentiation with multiplication) or make errors like \(\mathrm{4 \cdot 8 = 24}\), leading to wrong final answers. This may lead them to select Choice A (27) or Choice B (29) depending on where the error occurs.

Second Most Common Error:

Poor TRANSLATE reasoning: Misunderstanding what the problem is asking for

Some students might try to work backwards from the given table values or get confused about which variable represents what, leading to unnecessary complex approaches. This causes them to get stuck and guess among the answer choices.

The Bottom Line:

This problem tests whether students can systematically evaluate an exponential function using proper order of operations. The key is recognizing that it's a straightforward substitution problem, not requiring complex reasoning about the doubling pattern or table relationships.