A technician is monitoring the temperature of a chemical reaction. The temperature, T (in °C), at time t (in minutes) is given by the function T(t) = -t² + 10t + 20. What is the maximum temperature reached during the reaction?
- A70°C
- B25°C
- C45°C
- D20°C
Show answer & explanationAnswer & explanation
Correct answer: C. 45°C
The function T(t) = -t² + 10t + 20 is a quadratic function, representing a parabola opening downwards (since a = -1 < 0). The maximum temperature occurs at the vertex. First, find the time (t) at which the maximum occurs using t = -b/(2a). Here, a = -1 and b = 10, so t = -10 / (2 * -1) = -10 / -2 = 5 minutes. Then, substitute t = 5 back into the function to find the maximum temperature: T(5) = -(5)² + 10(5) + 20 = -25 + 50 + 20 = 45°C.
Why the other options are wrong
- A. This would be the result of incorrect calculation, perhaps using T(10) which is -100+100+20=20.
- B. This is the x-coordinate of the vertex (t=5) squared, but not the actual temperature.
- D. This is the initial temperature at t=0, not the maximum.
Finding Max/Min of Quadratic
For a quadratic function f(x) = ax² + bx + c, the maximum or minimum value occurs at the vertex. The x-coordinate of the vertex is -b/(2a). The y-coordinate (the max/min value) is found by plugging this x-coordinate back into the function.
- If 'a' is positive, the parabola opens upward, and the vertex is a minimum.
- If 'a' is negative, the parabola opens downward, and the vertex is a maximum.
- The vertex provides the extreme value (highest or lowest point) of the function.
Memory trick: Find the peak's location, then its height!