GED Mathematical Reasoning TestAlgebraic Problem Solving with Expressions and EquationsMedium
A technician is monitoring the temperature of a chemical reaction. The temperature, T, in degrees Celsius, is given by the function T(h) = -2h² + 16h + 20, where 'h' is the number of hours since the reaction started. What is the maximum temperature reached during the first 10 hours?
- A20°C
- B52°C
- C64°C
- D84°C
Show answer & explanationAnswer & explanation
Correct answer: B. 52°C
The function is a downward-opening parabola, so the maximum temperature occurs at the vertex. The h-coordinate of the vertex is -b/(2a) = -16/(2*-2) = -16/-4 = 4. Substitute h=4 into the function: T(4) = -2(4)² + 16(4) + 20 = -2(16) + 64 + 20 = -32 + 64 + 20 = 52. Since 4 hours is within the first 10 hours, this is the maximum.
Why the other options are wrong
- A. This is the initial temperature at h=0, not the maximum.
- C. This might come from an error in calculating the vertex or substituting the value.
- D. This could be a result of calculation errors or evaluating at an incorrect h-value.
Quadratic Function Maximum/Minimum
For a quadratic function f(x) = ax² + bx + c, the maximum or minimum value occurs at the vertex. If 'a' is negative, it's a maximum; if 'a' is positive, it's a minimum.
- Vertex x-coordinate: -b/(2a).
- Substitute x-coordinate into the function to find the max/min value.
- The sign of 'a' determines if the parabola opens up (min) or down (max).
Memory trick: Vertex is the peak or valley of your quadratic journey.