A quadratic function models the temperature, in degrees Fahrenheit, of a chemical reaction mixture over time, in minutes, after the...
GMAT Advanced Math : (Adv_Math) Questions
A quadratic function models the temperature, in degrees Fahrenheit, of a chemical reaction mixture over time, in minutes, after the reaction begins. The model indicates the mixture has an initial temperature of 67 degrees Fahrenheit and reaches its maximum temperature of 427 degrees Fahrenheit 6 minutes after the reaction begins. Based on the model, what is the temperature, in degrees Fahrenheit, of the mixture 8 minutes after the reaction begins?
1. TRANSLATE the problem information
- Given information:
- Quadratic function models temperature \(\mathrm{T(t)}\) over time \(\mathrm{t}\)
- Initial temperature: \(\mathrm{T(0) = 67°F}\)
- Maximum temperature: \(\mathrm{T(6) = 427°F}\)
- Need to find: \(\mathrm{T(8)}\)
- This tells us we need to determine the specific quadratic function, then evaluate it at \(\mathrm{t = 8}\).
2. INFER the approach
- Since we have a quadratic function, we can write \(\mathrm{T(t) = at² + bt + c}\)
- We need to find the three parameters \(\mathrm{a}\), \(\mathrm{b}\), and \(\mathrm{c}\)
- The key insight: the maximum occurs at \(\mathrm{t = 6}\), which means the vertex is at \(\mathrm{(6, 427)}\)
3. TRANSLATE initial condition
- From \(\mathrm{T(0) = 67}\): \(\mathrm{T(0) = a(0)² + b(0) + c = c}\)
- Therefore: \(\mathrm{c = 67}\)
4. INFER vertex relationship
- Since the maximum occurs at \(\mathrm{t = 6}\), this is the vertex of our parabola
- For quadratic \(\mathrm{at² + bt + c}\), the vertex occurs at \(\mathrm{t = -b/(2a)}\)
- Setting up: \(\mathrm{-b/(2a) = 6}\)
- SIMPLIFY: Cross-multiply to get \(\mathrm{b = -12a}\)
5. TRANSLATE maximum condition
- From \(\mathrm{T(6) = 427}\): \(\mathrm{427 = a(6)² + b(6) + 67}\)
- SIMPLIFY: \(\mathrm{427 = 36a + 6b + 67}\)
- Subtract 67: \(\mathrm{360 = 36a + 6b}\)
6. SIMPLIFY to solve for parameters
- Substitute \(\mathrm{b = -12a}\) into \(\mathrm{360 = 36a + 6b}\):
- \(\mathrm{360 = 36a + 6(-12a)}\)
- \(\mathrm{360 = 36a - 72a = -36a}\)
- Therefore: \(\mathrm{a = -10}\)
- And: \(\mathrm{b = -12(-10) = 120}\)
7. SIMPLIFY to evaluate \(\mathrm{T(8)}\)
- Our function: \(\mathrm{T(t) = -10t² + 120t + 67}\)
- \(\mathrm{T(8) = -10(64) + 120(8) + 67}\) (use calculator)
- \(\mathrm{T(8) = -640 + 960 + 67 = 387}\)
Answer: C (387)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students don't recognize that the maximum temperature occurs at the vertex of the parabola, so they don't use the vertex formula \(\mathrm{-b/(2a) = 6}\). Instead, they might try to set up a system using just the two given points \(\mathrm{(0, 67)}\) and \(\mathrm{(6, 427)}\), which gives them only two equations for three unknowns. This leads to confusion and guessing.
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly set up the equations but make algebraic errors when solving the system. For example, they might incorrectly substitute \(\mathrm{b = -12a}\) or make arithmetic errors when calculating \(\mathrm{T(8)}\). A common mistake is calculating \(\mathrm{T(8) = -10(8)² + 120(8) + 67 = -10(8) + 120(8) + 67 = 880 + 67 = 947}\), forgetting to square the 8 in the first term. This leads to confusion since none of the answer choices match such values.
The Bottom Line:
This problem tests whether students understand the connection between word problem constraints and the mathematical properties of quadratic functions. The critical insight is recognizing that "maximum temperature" translates to the vertex of a parabola, which then provides the key relationship needed to solve for all parameters.