Digital SATMath: AlgebraMedium
A projectile is launched vertically upward from the ground. Its height h, in meters, after t seconds, is given by the function h(t) = -5t² + 40t. At what time does the projectile reach its maximum height?
- A2 seconds
- B6 seconds
- C8 seconds
- D4 seconds
Show answer & explanationAnswer & explanation
Correct answer: D. 4 seconds
The function h(t) = -5t² + 40t is a quadratic function representing a downward-opening parabola. The maximum height occurs at the vertex of the parabola. The t-coordinate of the vertex can be found using the formula t = -b / (2a) for a quadratic equation at² + bt + c. Here, a = -5 and b = 40. So, t = -40 / (2 * -5) = -40 / -10 = 4 seconds.
Why the other options are wrong
- A. Incorrect. This would correspond to a different 'b' value in the vertex formula.
- B. Incorrect. This would correspond to a different 'a' value or calculation error.
- C. Incorrect. This is the time when the projectile returns to the ground (h(t)=0), not maximum height.
Vertex of a Parabola (Quadratic Function)
The highest or lowest point on the graph of a quadratic function, representing the maximum or minimum value of the function.
- For f(x) = ax² + bx + c, the x-coordinate of the vertex is -b/(2a).
- If a > 0, the parabola opens upward, and the vertex is a minimum.
- If a < 0, the parabola opens downward, and the vertex is a maximum.
- The y-coordinate is found by substituting the x-coordinate into the function.
Memory trick: Vertex is the top or bottom, -b/2a finds the spot.