ACT (Enhanced)MathematicsMedium
A meteorologist is tracking a weather balloon. The balloon's altitude (h) in meters above the ground, t minutes after launch, is given by the function h(t) = -t^2 + 16t + 20. What is the maximum altitude the weather balloon reaches?
- A20 meters
- B64 meters
- C128 meters
- D84 meters
Show answer & explanationAnswer & explanation
Correct answer: D. 84 meters
The function is a downward-opening parabola, so its maximum value occurs at the vertex. The t-coordinate of the vertex is -b/(2a), and the maximum altitude is h(-b/(2a)).
Why the other options are wrong
- A. This is the initial altitude (h(0)), not the maximum altitude.
- B. This is the square of the t-coordinate of the vertex, not the altitude itself.
- C. This results from a calculation error or misapplication of the vertex formula.
Vertex of a Parabola (Maximum)
For a quadratic function f(x) = ax^2 + bx + c with a < 0, the vertex represents the maximum value of the function.
- The x-coordinate of the vertex is given by x = -b / (2a).
- The maximum value is f(-b / (2a)).
- The parabola opens downwards when 'a' is negative.
Memory trick: Vertex's peak, -b/2a, then plug it back, no delay!