Basketball star mumford ( a seven foot senior forward ) places a mirror on the ground x ft from the vase of a basketball goal . he walks backwards six feet until he can see the top of the goal , which he knows is 10 feet tall . determine how far the mirror is from the basketball goal . justify your answer.

9 feet, 2 inches
   First thing to do is determine how high above the ground Mumford's eyes are. The proportions for an adult human is that their height is about 7.5 heads tall (this includes the head), so Mumford's head is about (7*12)/7.5 = 11.2 inches tall. The eyes are about centered vertically on the head, so Mumford's eyes are about 5.6 inches from the top of his head. So his eyes are about 7*12 - 5.6 = 84 - 5.6 = 78.4 inches from the ground. Once Mumford backs up and is able to see the goal in the mirror, he's created 2 similar right triangles. One of the triangles is the distance from the mirror to Mumford and the height of Mumford's eyes from the ground. The other triangle is the distance from the mirror to directly underneath the goal and the height of the goal. They're similar triangles since the angle of incidence of a light ray matches the angle of reflection. So let's do the math. First, the 6 feet and 10 feet measurements are multiplied by 12 to convert to inches. So we have 72 inches and 120 inches. Let's set up the equation and solve: 72/78.4 = X/120
 120*72/78.4 = X
 110.2040816 = X

   Let's round to the nearest inch giving 110 inches. And converting to feet and inches gives 9 feet, 2 inches.

Main idea on left, details on right