An investment account's value over time is modeled by the equation V = -2t^2 + 8t + 500, where V...
GMAT Advanced Math : (Adv_Math) Questions
An investment account's value over time is modeled by the equation \(\mathrm{V = -2t^2 + 8t + 500}\), where \(\mathrm{V}\) represents the account value in dollars and \(\mathrm{t}\) represents the number of years after the initial investment. Which number represents the initial amount invested in the account?
\(\mathrm{0}\)
\(\mathrm{2}\)
\(\mathrm{8}\)
\(\mathrm{500}\)
1. TRANSLATE the problem information
- Given information:
- Account value equation: \(\mathrm{V = -2t^2 + 8t + 500}\)
- \(\mathrm{V}\) = account value in dollars
- \(\mathrm{t}\) = years after initial investment
- Need to find: Initial amount invested
2. TRANSLATE what "initial amount" means mathematically
- "Initial amount invested" means the account value at the very beginning
- At the beginning, \(\mathrm{t = 0}\) (zero years after the initial investment)
- So we need to find \(\mathrm{V}\) when \(\mathrm{t = 0}\)
3. SIMPLIFY by substituting t = 0 into the equation
- \(\mathrm{V = -2t^2 + 8t + 500}\)
- \(\mathrm{V = -2(0)^2 + 8(0) + 500}\)
- \(\mathrm{V = -2(0) + 0 + 500}\)
- \(\mathrm{V = 0 + 0 + 500 = 500}\)
Answer: D (500)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students don't connect "initial amount invested" with evaluating the function at \(\mathrm{t = 0}\). Instead, they think the answer must be one of the coefficients in the equation since those are "the numbers in the problem."
They might reason: "The equation has -2, 8, and 500 as coefficients, so the initial amount must be one of these." This may lead them to select Choice C (8) or immediately jump to Choice D (500) without proper justification.
Second Most Common Error:
Conceptual confusion about variables: Students mix up which variable represents what, thinking they should substitute \(\mathrm{V = 0}\) instead of \(\mathrm{t = 0}\), or they don't understand that "initial" corresponds to the starting time value.
This leads to confusion and guessing among the answer choices.
The Bottom Line:
This problem tests whether students can connect real-world language ("initial amount") to mathematical operations (function evaluation at \(\mathrm{t = 0}\)). The key insight is recognizing that finding an initial value in a time-based model always means evaluating when the time variable equals zero.
\(\mathrm{0}\)
\(\mathrm{2}\)
\(\mathrm{8}\)
\(\mathrm{500}\)