The profit, P, in dollars, a company makes from selling an item for a price of p dollars can be...
GMAT Advanced Math : (Adv_Math) Questions
The profit, \(\mathrm{P}\), in dollars, a company makes from selling an item for a price of \(\mathrm{p}\) dollars can be modeled by a quadratic function. The company achieves its maximum daily profit of $128 when the selling price of the item is $20. The company makes a profit of $0 if the item is priced at $4. Which equation models the relationship between the company's profit and the item's price?
1. TRANSLATE the problem information
- Given information:
- Maximum daily profit: $128 when price = $20 → vertex at \(\mathrm{(20, 128)}\)
- Profit of $0 when price = $4 → point \(\mathrm{(4, 0)}\)
- Need to find quadratic function in form given by answer choices
2. INFER the approach
- Since we know the vertex (maximum point), vertex form is perfect: \(\mathrm{P = a(p - h)^2 + k}\)
- We have the vertex \(\mathrm{(h, k) = (20, 128)}\) but need to find coefficient 'a'
- The additional point \(\mathrm{(4, 0)}\) will help us find 'a'
3. TRANSLATE vertex into equation
- Substitute vertex coordinates into vertex form:
- \(\mathrm{P = a(p - 20)^2 + 128}\)
4. SIMPLIFY to find coefficient 'a'
- Use point \(\mathrm{(4, 0)}\): substitute \(\mathrm{p = 4}\) and \(\mathrm{P = 0}\)
- \(\mathrm{0 = a(4 - 20)^2 + 128}\)
- \(\mathrm{0 = a(-16)^2 + 128}\)
- \(\mathrm{0 = 256a + 128}\)
- \(\mathrm{-128 = 256a}\)
- \(\mathrm{a = -128/256 = -1/2}\)
5. INFER the final answer
- Substitute \(\mathrm{a = -1/2}\) back into the equation:
- \(\mathrm{P = -1/2(p - 20)^2 + 128}\)
Answer: A
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students may misinterpret "maximum profit of $128 when selling price is $20" and not recognize this describes the vertex coordinates \(\mathrm{(20, 128)}\). They might think the vertex is \(\mathrm{(128, 20)}\) or not understand what vertex means in this context.
This leads to using the wrong values in vertex form, causing them to select an incorrect answer or get confused and guess.
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly set up the equation \(\mathrm{0 = a(4-20)^2 + 128}\) but make algebraic errors. Common mistakes include:
- Calculating \(\mathrm{(4-20)^2}\) incorrectly as 16 instead of 256
- Sign errors when solving \(\mathrm{-128 = 256a}\)
- Fraction simplification errors when getting \(\mathrm{a = -1/2}\)
This may lead them to select Choice B (-2) if they get \(\mathrm{a = -2}\), or get stuck and guess.
The Bottom Line:
This problem requires translating business language into mathematical coordinates and then executing multi-step algebra accurately. Success depends on recognizing that vertex form connects directly to the "maximum profit" language.