Final answer:
The maximum height is determined using kinematic equations, setting the vertical velocity to zero at the apex of the trajectory to find the time and then solving for the height at that time.
Step-by-step explanation:
The goal is to find the maximum height a projectile can reach given the function h(t) = -16t² + v₀t + h₀, where v₀ is the initial velocity and h₀ is the initial height. By setting the vertical velocity to zero, we can find the time at which the projectile reaches its maximum height, since this is when it momentarily stops ascending before starting to fall back down. We can apply the kinematic equation v² = v₀² - 2g(y - y₀) to solve for the maximum height reached, where v is the final velocity (which is zero at maximum height), g is the acceleration due to gravity, y is the final position, and y₀ is the initial position.