Final answer:
To find the maximum height attained by the rocket, we calculate the vertex of the parabola represented by the function h(t) = -16t² + 192t + 806. By setting the derivative equal to zero, we find the time at which the maximum height occurs and substitute this into the original equation to find that the maximum height is 1214 feet.
Step-by-step explanation:
The question is about finding the maximum height attained by a rocket launched by a member of the local rocketry club. The rocket's height as a function of time is modeled by the function h(t) = −16t² + 192t + 806. To find the maximum height, we need to determine the vertex of this parabolic function, which represents the peak of the rocket's flight. The vertex form of a parabola is given by h(t) = a(t − h)² + k, where (h, k) is the vertex of the parabola. Since this is a downward-opening parabola (a is negative), the vertex represents the maximum point. The maximum height is determined by finding the value of t when the derivative h'(t) is zero, which indicates the peak of the parabola. Taking the derivative of h(t) and setting it to zero gives us the time at which the maximum height is reached. We can then substitute this value back into h(t) to find the maximum height the rocket achieves. The derivative of h(t) is h'(t) = −32t + 192. Setting h'(t) = 0 yields t = 6 seconds. Substituting this back into the original function, we find that the maximum height attained by the rocket is h(6) = −16(6)² + 192(6) + 806 = 1214 feet.