Final answer:
The rocket's greatest height is calculated using the vertex of the parabola represented by the quadratic equation h = -16t² + 32t + 84. By applying the formula for the vertex, the rocket's maximum height is found to be 100 feet.
Step-by-step explanation:
The student has provided the equation h = -16t² + 32t + 84 to determine the height of a toy rocket over time. To find the rocket's greatest height, we need to identify the vertex of the parabola represented by this quadratic equation. Since the coefficient of t² is negative, the parabola opens downwards, and the vertex will give us the maximum height.
The formula for the time at which the vertex occurs is given by t = -b/(2a), where 'a' is the coefficient of t² and 'b' is the coefficient of t. Substituting the values from the equation, we get t = -32/(2 * -16) = 1 second.
To find the greatest height, we substitute this value back into the original equation:
h = -16(1)² + 32(1) + 84 = -16 + 32 + 84 = 100 feet.
Therefore, the rocket's greatest height is 100 feet.