A company's weekly profit, in thousands of dollars, is modeled by the quadratic function P, where x is the selling...
GMAT Advanced Math : (Adv_Math) Questions
A company's weekly profit, in thousands of dollars, is modeled by the quadratic function \(\mathrm{P}\), where \(\mathrm{x}\) is the selling price, in dollars, of one of its products. The function \(\mathrm{P}\) estimates that the company achieves its maximum weekly profit of $405,000 when the selling price is $15 per product. The company makes no profit if the selling price is $30 per product. Based on the model, what is the estimated weekly profit, in thousands of dollars, if the selling price is $25 per product?
1. TRANSLATE the problem information
- Given information:
- Maximum weekly profit: \(\$405,000\) at selling price \(\$15\)
- Zero profit at selling price \(\$30\)
- Need profit when selling price is \(\$25\)
- Profit function \(\mathrm{P(x)}\) where \(\mathrm{x}\) = selling price, \(\mathrm{P(x)}\) = profit in thousands
- What this tells us:
- Vertex of parabola: \(\mathrm{(15, 405)}\) since profit is measured in thousands
- Point on curve: \(\mathrm{(30, 0)}\)
- Parabola opens downward (has maximum)
2. INFER the best approach
- Since we know the vertex and one other point, vertex form is most efficient
- Vertex form: \(\mathrm{P(x) = a(x - h)^2 + k}\) where \(\mathrm{(h,k)}\) is the vertex
- We need to find coefficient 'a' using the known point \(\mathrm{(30, 0)}\)
3. Set up the vertex form equation
\(\mathrm{P(x) = a(x - 15)^2 + 405}\)
4. SIMPLIFY to find coefficient 'a'
- Substitute point \(\mathrm{(30, 0)}\):
\(\mathrm{0 = a(30 - 15)^2 + 405}\)
\(\mathrm{0 = a(15)^2 + 405}\)
\(\mathrm{0 = 225a + 405}\)
\(\mathrm{-405 = 225a}\)
\(\mathrm{a = -405/225 = -1.8}\)
- Complete function: \(\mathrm{P(x) = -1.8(x - 15)^2 + 405}\)
5. SIMPLIFY to find P(25)
\(\mathrm{P(25) = -1.8(25 - 15)^2 + 405}\)
\(\mathrm{P(25) = -1.8(10)^2 + 405}\)
\(\mathrm{P(25) = -1.8(100) + 405}\)
\(\mathrm{P(25) = -180 + 405 = 225}\)
Answer: 225
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students may confuse the units, thinking the maximum profit of \(\$405,000\) means the vertex is \(\mathrm{(15, 405,000)}\) rather than \(\mathrm{(15, 405)}\).
When they use 405,000 in their vertex form and solve for 'a', they get \(\mathrm{a = -405,000/225 = -1,800}\). Then calculating P(25) gives them \(\mathrm{-180,000 + 405,000 = 225,000}\). This leads to confusion about whether the answer should be 225 or 225,000, potentially causing them to guess incorrectly.
Second Most Common Error:
Poor INFER reasoning: Students might attempt to use standard form \(\mathrm{ax^2 + bx + c}\) instead of recognizing that vertex form is more efficient given the vertex information.
They may try to set up a system of equations using the three known points, leading to unnecessarily complex algebra. This increases chances of computational errors and may cause them to abandon the systematic solution and guess.
The Bottom Line:
This problem tests whether students can effectively translate contextual information into the right mathematical form and choose an efficient solution strategy. The key insight is recognizing that vertex form directly uses the given maximum point information.