Final answer:
The equation to describe the height of a flare over time, when shot from 40 feet below ground level at 65 feet per second, is h(t) = -16t^2 + 65t - 40.
Step-by-step explanation:
The equation to represent the relationship between the height of the flare and time when a man shoots a flare straight up from a point 40 feet below ground level with an initial velocity of 65 feet per second is given by the quadratic equation derived from the laws of physics describing projectile motion. The equation will take into account the initial velocity, the acceleration due to gravity, and the initial height from which the flare is launched.
The standard form of the equation is h(t) = -16t2 + vt + s, where h(t) is the height of the flare at time t, v is the initial velocity, and s is the initial height. Since we know the initial velocity (v) is 65 feet per second and the initial height (s) is -40 feet (because it's below ground level), the equation for this situation is h(t) = -16t2 + 65t - 40.