A company opens an account with an initial balance of $36,100.00. The account earns interest, and no additional deposits or...

1. TRANSLATE the problem information

• Given information:
  • Initial balance: $36,100.00 (this is when t = 0)
  • Balance after 13 years: $68,071.93 (this is when t = 13)
  • Function follows exponential form A(t)

2. INFER the exponential function structure

• An exponential function has the form \(\mathrm{A(t) = N \times r^t}\) where:
  • \(\mathrm{N}\) = initial value (when t = 0)
  • \(\mathrm{r}\) = growth factor
• Since the initial balance is $36,100.00, we know \(\mathrm{N = 36,100.00}\)
• We can verify: \(\mathrm{A(0) = 36,100.00 \times r^0 = 36,100.00 \times 1 = 36,100.00}\) ✓

3. TRANSLATE the 13-year condition into an equation

• We know \(\mathrm{A(13) = \$68,071.93}\)
• Substituting into our function: \(\mathrm{36,100.00 \times r^{13} = 68,071.93}\)

4. SIMPLIFY to find the growth factor r

• Divide both sides by 36,100.00:
  \(\mathrm{r^{13} = 68,071.93 \div 36,100.00 = 1.885}\) (use calculator)
• Take the 13th root of both sides:
  \(\mathrm{r = (1.885)^{1/13} \approx 1.05}\) (use calculator)

5. INFER the complete function

• Substituting our values: \(\mathrm{A(t) = 36,100.00(1.05)^t}\)
• This matches choice A exactly

Answer: A. \(\mathrm{A(t) = 36,100.00(1.05)^t}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't understand the structure of exponential functions and confuse which value represents the initial amount versus the growth factor. They might think that since 0.05 represents 5%, choice D with \(\mathrm{A(t) = 36,100.00(0.05)^t}\) looks reasonable. However, \(\mathrm{0.05^t}\) represents exponential decay (getting smaller each year), not growth. The correct growth factor for 5% annual growth is 1.05, not 0.05.

This may lead them to select Choice D (\(\mathrm{36,100.00(0.05)^t}\))

Second Most Common Error:

Poor TRANSLATE reasoning: Students might incorrectly identify which dollar amount represents the initial value N. They could confuse the setup and think the final amount ($68,071.93) minus some amount gives the initial coefficient, leading them toward choice B with the wrong initial value of $31,971.93.

This may lead them to select Choice B (\(\mathrm{31,971.93(1.05)^t}\))

The Bottom Line:

This problem requires students to deeply understand exponential function structure and correctly match real-world information to mathematical parameters. The key insight is recognizing that 1.05 represents 5% growth (not 0.05) and that the initial balance directly becomes the coefficient N in the exponential function.