Digital SATMath: Advanced MathMedium
A rocket engineer is calculating the maximum height achieved by a test rocket. The height, h(t), in meters, at time t seconds after launch, is given by the function h(t) = -4.9t^2 + 196t + 10. What is the maximum height the rocket reaches?
- A10 meters
- B20 seconds
- C1,970 meters
- D2,010 meters
Show answer & explanationAnswer & explanation
Correct answer: D. 2,010 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). For h(t) = -4.9t^2 + 196t + 10, a = -4.9 and b = 196. So, t = -196 / (2 * -4.9) = -196 / -9.8 = 20 seconds. Substitute t=20 into the function: h(20) = -4.9(20)^2 + 196(20) + 10 = -4.9(400) + 3920 + 10 = -1960 + 3920 + 10 = 1970 + 10 = 2010 meters.
Why the other options are wrong
- A. This is the initial height of the rocket at t=0.
- B. This is the time at which the maximum height is reached, not the maximum height itself.
- C. This is a calculation error, likely from -1960 + 3920 = 1960, and then adding 10 to get 1970.
Vertex of a Parabola
The highest or lowest point on the graph of a quadratic function, representing the maximum or minimum value of the function.
- For y = ax^2 + 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.
Memory trick: Vertex is the 'peak' or 'pit' of the parabola.