ACT (Enhanced)MathematicsMedium
A meteorologist is tracking a weather balloon. The balloon's altitude (h) in meters above ground after t minutes is given by the function h(t) = -t^2 + 10t + 50. What is the maximum altitude the balloon reaches?
- A50 meters
- B100 meters
- C55 meters
- D75 meters
Show answer & explanationAnswer & explanation
Correct answer: D. 75 meters
The function h(t) = -t^2 + 10t + 50 is a downward-opening parabola. The maximum altitude corresponds to the y-coordinate of the parabola's vertex. The x-coordinate (time) of the vertex is found using t = -b / (2a).
Why the other options are wrong
- A. Incorrect. This is the initial altitude (h(0)), not the maximum altitude.
- B. Incorrect. This would be the altitude if the coefficient of t^2 was positive, or if there was a calculation error.
- C. Incorrect. This would be the altitude if the time to reach max altitude was 2.5 minutes (e.g., h(2.5) = -6.25 + 25 + 50 = 68.75 or other calculation error).
Vertex of a Parabola (Maximum/Minimum)
For a quadratic function in the form f(x) = ax^2 + bx + c, the vertex represents the maximum point if a < 0 (parabola opens down) or the minimum point if a > 0 (parabola opens up).
- x-coordinate of vertex: -b / (2a).
- y-coordinate of vertex: substitute x-coordinate into the function.
- If 'a' is negative, the vertex is a maximum; if 'a' is positive, it's a minimum.
Memory trick: The 'Vertex' is the 'Peak' or 'Valley' of the 'Parabola's Path'.