Final answer:
To find the maximum height reached by the rocket from the given quadratic equation, calculate the vertex of the parabola. The x-coordinate of the vertex represents the time to reach the maximum height, and substituting this back into the equation gives the y-coordinate, which is the maximum height of approximately 918.4 feet.
Step-by-step explanation:
Finding the Maximum Height of a Rocket
To find the maximum height reached by the rocket, we must analyze the quadratic equation that describes its motion: y = -16x² + 242x + 73. The equation represents a parabola opening downwards (since the coefficient in front of x² is negative), which implies that the vertex of the parabola is the point where the rocket reaches its maximum height.
The horizontal coordinate of the vertex can be found using the formula x = -b/(2a), where a is the coefficient in front of x², and b is the coefficient in front of x. For the given equation, a is -16 and b is 242. Plugging these values into the formula gives us x = -242/(2 × -16) ≈ 7.5625 seconds.
To find the maximum height, which is the vertical coordinate of the vertex, we substitute x = 7.5625 back into the original quadratic equation, yielding the maximum height y ≈ -16(7.5625)² + 242(7.5625) + 73. Calculating this gives us a maximum height of approximately 918.4 feet, to the nearest tenth of a foot.
This process of finding the max height is consistent with the motion equations encountered in projectile motion, where the maximum height is influenced by the initial vertical velocity and the downward acceleration due to gravity. Note that the maximum height can be derived without knowing the final velocity at the max height, as it is momentarily zero (vy = 0).