ACT (Enhanced)MathematicsHard
A meteorologist is tracking a weather balloon. The balloon's altitude (h) in meters above ground is given by the function h(t) = -2t² + 40t + 50, where 't' is the time in minutes since launch. What is the maximum altitude the balloon reaches?
- A50 meters
- B250 meters
- C200 meters
- D450 meters
Show answer & explanationAnswer & explanation
Correct answer: B. 250 meters
The function h(t) is a downward-opening parabola (because the coefficient of t² is negative). The maximum altitude is at the vertex. The t-coordinate of the vertex is given by t = -b / (2a). For h(t) = -2t² + 40t + 50, a = -2 and b = 40. So, t = -40 / (2 * -2) = -40 / -4 = 10 minutes. Substitute t=10 back into h(t): h(10) = -2(10)² + 40(10) + 50 = -2(100) + 400 + 50 = -200 + 400 + 50 = 250 meters.
Why the other options are wrong
- A. This is the initial altitude at t=0, not the maximum.
- C. This would be the result if 50 was not added at the end of the calculation, or an error in the vertex formula.
- D. This is an incorrect calculation, possibly if 'a' was positive or a sign error occurred.
Vertex of a Parabola (Quadratic)
The highest or lowest point on the graph of a quadratic function (parabola), representing the maximum or minimum value of the function.
- For f(x) = ax² + bx + c, the x-coordinate of the vertex is -b/(2a).
- Substitute the x-coordinate back into the function to find the y-coordinate (max/min value).
- If a > 0, parabola opens up (minimum); if a < 0, parabola opens down (maximum).
Memory trick: Vertex Peak: X is minus B over 2A, then plug X back in!