\(\mathrm{V(h) = 5,000(1.08)^h}\) The given function V models the value, in dollars, of an investment h hours after it was...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{V(h) = 5,000(1.08)^h}\) models investment value after h hours
  • Need to find \(\mathrm{W(n)}\) that models value after n minutes

• What this tells us: We need to convert between hours and minutes in our function

2. INFER the conversion relationship

• Key insight: Since we're changing from hours to minutes, we need to express h in terms of n
• Since \(\mathrm{1\ hour = 60\ minutes}\), if the investment has been made for n minutes, that equals \(\mathrm{n/60}\) hours
• Therefore: \(\mathrm{h = n/60}\)

3. SIMPLIFY by substitution

• Replace h with \(\mathrm{n/60}\) in the original function:
  \(\mathrm{V(h) = 5,000(1.08)^h}\) becomes \(\mathrm{V(n/60) = 5,000(1.08)^{(n/60)}}\)
• This gives us our new function: \(\mathrm{W(n) = 5,000(1.08)^{(n/60)}}\)

Answer: C. \(\mathrm{W(n) = 5,000(1.08)^{(n/60)}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students often confuse which direction the conversion should go. They might think "since there are 60 minutes in an hour, I multiply by 60" and incorrectly conclude \(\mathrm{h = 60n}\) instead of \(\mathrm{h = n/60}\).

This backwards relationship leads them to select Choice D (\(\mathrm{W(n) = 5,000(1.08)^{(60n)}}\)).

Second Most Common Error:

Incomplete TRANSLATE understanding: Students recognize they need to involve "60" somewhere but incorrectly modify other parts of the function instead of just the exponent. They might divide the initial value by 60 or modify the base.

This may lead them to select Choice A (\(\mathrm{W(n) = (5,000/60)(1.08)^n}\)) or Choice B (\(\mathrm{W(n) = 5,000(1.08/60)^n}\)).

The Bottom Line:

The key insight is recognizing that when converting time units in the input variable of a function, you must substitute the conversion relationship directly into the function. The conversion factor (60) affects how the time variable appears in the exponent, not the base or coefficient of the exponential function.