A person deposits an initial amount of $5,000 into a savings account. The account earns a 4% annual interest rate,...

1. TRANSLATE the problem information

• Given information:
  • Initial deposit: \(\$5,000\)
  • Annual interest rate: \(4\%\)
  • Compounded annually
  • Time variable: \(\mathrm{t}\) years
  • No additional deposits or withdrawals

• What we need: A function \(\mathrm{A(t)}\) for account balance after \(\mathrm{t}\) years

2. INFER the mathematical model needed

• Since the problem mentions "compounded annually," this means the account balance grows exponentially, not linearly
• Each year, the entire balance (not just the original deposit) earns interest
• This requires the compound interest formula: \(\mathrm{A = P(1 + r)^t}\)

3. TRANSLATE the values into the formula

• \(\mathrm{P}\) (principal) = \(5,000\)
• \(\mathrm{r}\) (interest rate as decimal) = \(4\% = 0.04\)
• The growth factor is \(\mathrm{(1 + r) = 1 + 0.04 = 1.04}\)

4. Assemble the complete function

• \(\mathrm{A(t) = P(1 + r)^t}\)
• \(\mathrm{A(t) = 5,000(1.04)^t}\)

Answer: D. \(\mathrm{A(t) = 5,000(1.04)^t}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about compound vs. simple interest: Students may not recognize that "compounded annually" means exponential growth where the entire balance earns interest each year, not just the original principal.

Instead, they think of simple interest where only the original amount earns interest each year, leading them to use a linear model: \(\mathrm{A(t) = 5,000(1 + 0.04t)}\). This may lead them to select Choice C (\(\mathrm{A(t) = 5,000(1 + 0.04t)}\)).

Second Most Common Error:

Weak TRANSLATE skill with percentage conversion: Students might incorrectly interpret "4% growth" as meaning the balance gets multiplied by 1.4 each year instead of 1.04, perhaps thinking 4% means adding 0.4 instead of 0.04.

This may lead them to select Choice B (\(\mathrm{A(t) = 5,000(1.4)^t}\)).

The Bottom Line:

The key challenge is recognizing that compound interest creates exponential growth (where the growth factor is 1 + r) and distinguishing this from simple interest's linear growth pattern.