Very confused need process and solution

  33 ft

Step-by-step explanation:

The image in a mirror is as far behind the reflecting surface as the actual object is in front. When working reflection problems such as this one, it is convenient to mirror the geometry in the line of reflection. This puts the target point at the same distance from the left edge of the room, but 8 feet beyond the reflecting wall. A straight line from the source to the target image will intersect the wall at the appropriate point.

Essentially, we want to know where a line between a point at coordinates (6, 12) and a point at coordinates (45, 38) will intersect the wall at y=30.

The 2-point form of the equation for the line is ...

  y = (y2 -y1)/(x2 -x1)(x -x1) +y1

For the points listed above, this is ...

  y = (38 -12)/(45 -6)(x -6) +12 = 26/39(x -6) +12

  y = 2/3x +8

The value of x that gives y=30 is ...

  30 = 2/3x +8 . . . . . substitute 30 for y

  22 = 2/3x . . . . . . . . subtract 8

  (3/2)·22 = x = 33 . . . . multiply by 3/2

Zack should shoot at a point 33 feet along the wall in order to hit Patrick.

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In short, Zack should aim at Patrick's reflection in the wall.