A company's daily profit, P, in thousands of dollars, is related to its daily advertising spending, x, in thousands of...
GMAT Advanced Math : (Adv_Math) Questions
A company's daily profit, P, in thousands of dollars, is related to its daily advertising spending, x, in thousands of dollars. The relationship is modeled by the equation \(\mathrm{P} = -2(\mathrm{x} - 9)^2 + 162\). Which of the following is the best interpretation of the vertex of the graph of this equation?
1. TRANSLATE the equation structure
- Given: \(\mathrm{P = -2(x - 9)^2 + 162}\)
- This matches vertex form: \(\mathrm{P = a(x - h)^2 + k}\)
- We can identify: \(\mathrm{a = -2, h = 9, k = 162}\)
2. INFER what type of extremum this represents
- Since \(\mathrm{a = -2 \lt 0}\), the parabola opens downward
- This means we have a maximum, not a minimum
- The vertex represents the highest point on the graph
3. TRANSLATE the vertex coordinates to context
- The vertex occurs at \(\mathrm{(h, k) = (9, 162)}\)
- In context:
- \(\mathrm{x = 9}\) represents $9,000 in daily advertising spending (since x is in thousands)
- \(\mathrm{P = 162}\) represents $162,000 in daily profit (since P is in thousands)
4. TRANSLATE this into the answer format
- Maximum daily profit: $162,000
- Advertising spending that achieves this: $9,000
- This matches choice A exactly
Answer: A
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students correctly identify the vertex as (9, 162) but swap which value represents what in their interpretation.
They might think: "The vertex is (9, 162), so maximum profit is $9,000 with advertising spending of $162,000." This reverses the roles of x and P, leading them to select Choice B ($9,000 profit, $162,000 advertising).
Second Most Common Error:
Poor TRANSLATE reasoning: Students see the coefficient -2 in the equation and incorrectly use it as either the profit amount or advertising amount, instead of recognizing it only affects the parabola's shape.
They might select Choice C ($2,000 profit) or Choice D ($2,000 advertising) by mistakenly incorporating the coefficient -2 into their final answer.
The Bottom Line:
Success requires carefully matching mathematical coordinates to their real-world meanings. The vertex coordinates tell the complete story - you just need to translate them correctly using the given context and units.