ACT (Enhanced)MathematicsMedium
A drone operator is programming a flight path. The drone's altitude (h) in meters above the ground is modeled by the function h(t) = -5t² + 40t + 10, where 't' is the time in seconds after takeoff. What is the maximum altitude the drone reaches?
- A10 meters
- B90 meters
- C40 meters
- D80 meters
Show answer & explanationAnswer & explanation
Correct answer: B. 90 meters
The function is a downward-opening parabola (because of -5t²). The maximum altitude is at the vertex. The t-coordinate of the vertex is -b/(2a), which is -40 / (2 * -5) = -40 / -10 = 4 seconds. Substitute t=4 into the function: h(4) = -5(4)² + 40(4) + 10 = -5(16) + 160 + 10 = -80 + 160 + 10 = 90 meters.
Why the other options are wrong
- A. This is the initial altitude (h(0)).
- C. This is a coefficient, not the maximum altitude.
- D. Incorrect calculation, possibly misinterpreting the vertex formula or substitution.
Vertex of a Parabola
For a quadratic function f(x) = ax² + bx + c, the vertex is the highest or lowest point on the parabola. Its x-coordinate is given by -b/(2a).
- If 'a' is positive, vertex is a minimum.
- If 'a' is negative, vertex is a maximum.
- The y-coordinate of the vertex gives the maximum/minimum value.
Memory trick: Quadratic peaks or valleys, like a rollercoaster's highest point.