Distance from mirror
A dressing mirror mounted on a vertical wall is tall with the bottom above the floor. A bare light-bulb hangs on the ceiling with a horizontal distance of away from the wall with the mirror. The light-bulb is above the floor. Approximately how long is the streak of the reflected light across the floor, as measured perpendicularly away from the wall with the mirror?
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1 solution
The light hitting the top of the mirror will reflect the furthest and the light hitting the bittom of the mirror will end up closest to the wall. The top of the mirror is 1.7+0.3=2m above the floor. The lightbulb is 3m off the floor so is therefore 1m higher. We are given that the lightbulb is 1m away from the mirror so using this information we can picture a right angled triangle of which the height is the vertical distance between the top of the mirror and the lightbulb and the base is the horizontal distance. The other angles in the right angled isosceles triangle are 90° so the angle of incidence is 45° and so is the angle of reflection. We can then construct another right angled triangle in which the height is 2m (the distance to the floor) and the angles are 45°,45° and 90°. This means that the width of the triangle is 2m because the triangle is isosceles so the furthest ray of light will hit the floor 2m away from the wall. Using similar principles as above, we can calculate that the angle of incidence for the bottom of the mirror is 69.67686...° because the height of the right angled triangle is 2.7m and the width is 1m and then we can use SOHCAHTOA. This means that the angle of reflection is also 69.67686...° so the angle between the ray of light and the wall is 90 - 69.67686= 20.32314°. The distance above the ground is 0.3m so tan(20.32314) x 0.3 = 0.111..m so the closest ray is approximately 0.1m away from the wall. 2m - 0.1m = 1.9m therefore the streak of reflected light is about 1.9m long.