The function F is defined by \(\mathrm{F(t) = 250(1.6)^t}\). A new function G is defined by \(\mathrm{G(t) = F(t +...

1. TRANSLATE the function relationship

• Given information:
  • \(\mathrm{F(t) = 250(1.6)^t}\)
  • \(\mathrm{G(t) = F(t + 3)}\)
• This means wherever we see t in F, we replace it with \(\mathrm{(t + 3)}\)

2. TRANSLATE the substitution into mathematical form

• \(\mathrm{G(t) = F(t + 3) = 250(1.6)^{(t+3)}}\)
• Now we need to rewrite this in the form that matches the answer choices

3. INFER the strategy for simplification

• The answer choices all have the form: \(\mathrm{coefficient \times (1.6)^t}\)
• We need to use exponent properties to separate \(\mathrm{(1.6)^{(t+3)}}\) into parts

4. SIMPLIFY using exponent properties

• Apply the rule \(\mathrm{a^{(m+n)} = a^m \cdot a^n}\):

\(\mathrm{G(t) = 250(1.6)^{(t+3)}}\)
\(\mathrm{G(t) = 250(1.6)^t \cdot (1.6)^3}\)

• This separates the expression into the desired form

5. SIMPLIFY the coefficient calculation

• Calculate \(\mathrm{(1.6)^3}\) step by step:

\(\mathrm{(1.6)^2 = 2.56}\)
\(\mathrm{(1.6)^3 = 2.56 \times 1.6 = 4.096}\) (use calculator)

• Find the new coefficient:

\(\mathrm{250 \times 4.096 = 1024}\) (use calculator)

6. Write the final form

• \(\mathrm{G(t) = 1024(1.6)^t}\)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misunderstand what \(\mathrm{G(t) = F(t + 3)}\) means and try to add 3 to the entire function instead of substituting \(\mathrm{(t + 3)}\) for t.

This leads them to write something like \(\mathrm{G(t) = 250(1.6)^t + 3}\) or \(\mathrm{G(t) = 253(1.6)^t}\), which doesn't match any answer choice and causes confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{G(t) = 250(1.6)^{(t+3)}}\) but make arithmetic errors when calculating \(\mathrm{(1.6)^3}\) or the final multiplication.

Common calculation mistakes include getting \(\mathrm{(1.6)^3 = 2.56}\) (forgetting the third power) or making multiplication errors, leading them to select Choice A (640), Choice B (400), or Choice C (625).

The Bottom Line:

This problem tests whether students can correctly interpret function composition notation and then execute the algebraic manipulation systematically. The key insight is recognizing that function transformation requires substitution, not simple addition.