A tech startup's monthly revenue can be modeled by the equation R = -2.5m^2 + 15m + 24, where R...
GMAT Advanced Math : (Adv_Math) Questions
A tech startup's monthly revenue can be modeled by the equation \(\mathrm{R = -2.5m^2 + 15m + 24}\), where \(\mathrm{R}\) represents the revenue in thousands of dollars and \(\mathrm{m}\) represents the number of months after the company began operations. According to this model, what was the company's revenue, in thousands of dollars, when the company began operations?
Enter your answer as an integer.
1. TRANSLATE the problem information
- Given information:
- Revenue equation: \(\mathrm{R = -2.5m^2 + 15m + 24}\)
- \(\mathrm{R}\) = revenue in thousands of dollars
- \(\mathrm{m}\) = months after company began operations
- Need to find: revenue when company began operations
- What this tells us: "When the company began operations" means at the very start, which corresponds to \(\mathrm{m = 0}\) (zero months after beginning).
2. SIMPLIFY by substituting and calculating
- Substitute \(\mathrm{m = 0}\) into the equation:
\(\mathrm{R = -2.5(0)^2 + 15(0) + 24}\)
- Apply order of operations:
- First: \(\mathrm{(0)^2 = 0}\)
- Then: \(\mathrm{-2.5(0) = 0}\) and \(\mathrm{15(0) = 0}\)
- Finally: \(\mathrm{R = 0 + 0 + 24 = 24}\)
Answer: 24
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students may not recognize that "when the company began operations" corresponds to \(\mathrm{m = 0}\). They might think this means \(\mathrm{m = 1}\) (first month) or get confused about what the starting point represents in the context.
This leads to substituting the wrong value for m, such as \(\mathrm{m = 1}\), which would give \(\mathrm{R = -2.5(1)^2 + 15(1) + 24 = -2.5 + 15 + 24 = 36.5}\). This leads to confusion and an incorrect answer.
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly identify that \(\mathrm{m = 0}\), but make arithmetic errors during substitution, such as forgetting that anything multiplied by 0 equals 0, or making order of operations mistakes.
This causes calculation errors and leads to selecting an incorrect numerical answer or getting stuck.
The Bottom Line:
The key insight is recognizing that "began operations" refers to the starting point (\(\mathrm{m = 0}\)) in the timeline, not the first month of operations. Once this translation is made correctly, the problem becomes a straightforward substitution and arithmetic exercise.