An investment account was opened with an initial value of $890. The value of the account doubled every 10 years....

1. TRANSLATE the problem information

• Given information:
  • Initial investment value: \(\$890\)
  • The value doubles every 10 years
  • Need equation for \(\mathrm{M(t)}\) where \(\mathrm{t}\) = years after opening

• What this tells us: This is exponential growth with a specific doubling period

2. INFER the exponential growth structure

• Since the value doubles at regular intervals, this follows the exponential growth pattern
• We need the form: \(\mathrm{M(t)}\) = Initial value × (growth factor)^{(number of growth periods)}
• The key insight: We need to figure out how many 10-year periods occur in \(\mathrm{t}\) years

3. TRANSLATE the doubling pattern into mathematical terms

• "Doubles every 10 years" means growth factor = 2
• Number of 10-year periods in \(\mathrm{t}\) years = \(\mathrm{t/10}\)
• So the exponent should be \(\mathrm{t/10}\), not just \(\mathrm{t}\)

4. Construct the equation

• \(\mathrm{M(t)} = 890 \times 2^{(t/10)}\)
• This can be written as \(\mathrm{M(t)} = 890(2)^{(t/10)}}\)

5. Verify by checking the pattern

• At \(\mathrm{t} = 0\):
  \(\mathrm{M(0)} = 890(2)^{(0/10)}\)
  \(= 890(2)^0\)
  \(= 890 \times 1\)
  \(= \$890\) ✓

• At \(\mathrm{t} = 10\):
  \(\mathrm{M(10)} = 890(2)^{(10/10)}\)
  \(= 890(2)^1\)
  \(= 890 \times 2\)
  \(= \$1,780\) ✓

• At \(\mathrm{t} = 20\):
  \(\mathrm{M(20)} = 890(2)^{(20/10)}\)
  \(= 890(2)^2\)
  \(= 890 \times 4\)
  \(= \$3,560\) ✓

Answer: C. \(\mathrm{M(t)} = 890(2)^{(t/10)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students correctly identify that doubling means using base 2, but mishandle the time period relationship. They might think "since t represents years, I just use t as the exponent" and create \(\mathrm{M(t)} = 890(2)^t\). This ignores that doubling happens every 10 years, not every year.

This fundamental misunderstanding of the time scaling leads to confusion when checking answer choices, typically resulting in guessing between the remaining options.

Second Most Common Error:

Conceptual confusion about growth direction: Students may confuse "doubling" with "halving" or get confused about growth vs decay. They might see 1/2 in choice A and think this represents doubling, leading them to select Choice A (\(\mathrm{M(t)} = 890(1/2)^{(t/10)}\)) which actually represents the value halving every 10 years.

The Bottom Line:

The key challenge is correctly TRANSLATING the phrase "doubles every 10 years" into the mathematical relationship \(\mathrm{t/10}\) in the exponent. Students who miss this time-scaling concept cannot construct the correct exponential model.