The function \(\mathrm{V(t) = 1{,}200(4)^{(t/18)}}\) models the value, in dollars, of a rare collectible t years after it was purchased....

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{V(t) = 1{,}200(4)^{(t/18)}}\) represents the value in dollars after t years
  • Need to find when value doubles

• What this tells us: We need to find t when \(\mathrm{V(t) = 2 \times}\) initial value

2. TRANSLATE to find the target value

• Initial value at \(\mathrm{t = 0}\): \(\mathrm{V(0) = 1{,}200(4)^{(0/18)} = 1{,}200}\)
• Doubled value = \(\mathrm{2 \times 1{,}200 = 2{,}400}\)

• Set up the equation: \(\mathrm{2{,}400 = 1{,}200(4)^{(t/18)}}\)

3. SIMPLIFY the equation setup

• Divide both sides by 1,200: \(\mathrm{2 = (4)^{(t/18)}}\)
• Now we have a clean exponential equation to solve

4. INFER the solution strategy

• To solve exponential equations, we need equal bases
• Since \(\mathrm{4 = 2^2}\), we can rewrite: \(\mathrm{2 = (2^2)^{(t/18)}}\)
• This gives us: \(\mathrm{2^1 = 2^{(2t/18)}}\)

5. SIMPLIFY to solve for t

• With equal bases, set exponents equal: \(\mathrm{1 = 2t/18}\)
• Multiply both sides by 18: \(\mathrm{18 = 2t}\)
• Divide by 2: \(\mathrm{t = 9}\)

Answer: B. 9

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may misunderstand what "double" means in context and set up \(\mathrm{V(t) = 2V(t)}\) instead of \(\mathrm{V(t) = 2 \times V(0)}\).

This fundamental misunderstanding leads to confusion because they're trying to solve an impossible equation. This leads to abandoning systematic solution and guessing.

Second Most Common Error:

Poor INFER reasoning: Students recognize they need to solve \(\mathrm{2 = (4)^{(t/18)}}\) but don't know the strategy of making bases equal. They might try to take logarithms without proper technique or attempt trial-and-error with the answer choices.

Without the key insight of converting \(\mathrm{4}\) to \(\mathrm{2^2}\), students get stuck in the exponential equation. This may lead them to select Choice C (18) by incorrectly thinking it relates to the denominator in the exponent.

The Bottom Line:

This problem tests whether students can set up exponential equations from word problems and solve them using the fundamental strategy of equal bases. The "doubling" language and the base conversion (\(\mathrm{4 = 2^2}\)) are the two critical hurdles that determine success.