A water balloon was launched into the air from the top of a hill 38 feet above ground. The balloon's path is represented by the equation
h(t) = -162 + 48t + 38, where h(t) represents the height, in feet, and t is the time in seconds. Find the
average rate of change from the initial launch to the maximum height.

let's first notice, the center of the hyperbola is at the origin, since the focus points are equidistantly 3 units away from the origin, that also means that the distance "c" from the center to either focus is 3.

it should also be notable that, the traverse axis for this hyperbola is vertical, or namely over the y-axis and therefore, the positive fraction will be the one with y-variable.

the traverse "a" component or distance from the center to either vertex is not 2, is more than 2, so we can safely put it at √5 ≈ 2.24.

it would be 3 0ver 24 very unlikely.