Final answer:
Using the quadratic function modeling the height of the rock over time, it takes 0.5 seconds for the rock to reach its maximum height. This is found by using the vertex formula on the given function.
Step-by-step explanation:
To determine how long it took for the rock to reach maximum height, we use the modeled function for the height of the rock as a function of time, which is h(t) = -16t² + 16t + 32. This is a quadratic equation that represents the path of a projectile under gravity, where -16t² represents the acceleration due to gravity (in feet per second squared), 16t is the initial vertical velocity component, and 32 is the initial height. For a quadratic function in the form of ax² + bx + c, the vertex, or the maximum point, occurs at t = -b/(2a). For our given function, a = -16 and b = 16. Substituting these values into the vertex formula gives us: t = -16 / (2 · -16) = 0.5 seconds. Therefore, the rock reaches its maximum height 0.5 seconds after it is thrown. Solution summary The maximum height the rock will reach can be found by using the vertex of the parabola represented by the quadratic function. By applying the vertex formula to the coefficients of the given function, we calculate that it takes 0.5 seconds for the rock to reach its maximum height.