How high must a 48-inch high mirror be hung if one who is six feet tall needs to see his entire image from approximately 6 feet away from the mirror?
No matter what the horizontal distance between the user and the mirror, the bottom edge of the mirror must be no higher than half the height of the user's eyes. If so, the ray of vision starting from the user's eyes and bouncing off the mirror right at the bottom edge will hit the ground just where his feet are. The horizontal distance is not important because the mirror is always exactly halfway between you and your image behind the looking-glass.
Since 48" is 121 cm, well above half of most people's height, there's a range of positions that will suffice, and you can choose among those by aesthetic preferences. I think it would would probably do well to hang the mirror such that its top edge is about a hand's width above the top of the tallest user's head. Unless that will make the bottom so high that the shortest users will have difficulty seeing themselves completely -- in which case lower it, but not more than such that the top edge is a few centimeters below the top of the tallest user's head.