Question

# A person's eyes are h = 1.7 m above the floor as he stands d = 2.5 m away from a vertical plane mirror. The bottom edge of the mirror is at a height of y above the floor.a. The person looks at the bottom edge of the mirror and sees a reflection from points on the floor that are x = 0.55 m horizontally away from the mirror. How high, in meters, is the bottom edge of the mirror above the floor?b. If the person doubles his distance from the mirror, what horizontal distance, in meters, along the floor from the mirror will he see when he looks at the bottom edge of the mirror?

Answer

a) 2.15 m, b) -0.424 m

- Identify the given values: h = 1.7 m, d = 2.5 m, y = unknown, x = 0.55 m
- Use the law of reflection to determine the height of y. In this case, the angle of incidence (i) is equal to the angle of reflection (r). tan(i) = x/d, therefore, i = arctan(x/d). The angle of incidence also satisfies the relation i = arctan((y-h)/d). Therefore, we have arctan((y-h)/d) = arctan(x/d), which simplifies to y = h + d*tan(arctan(x/d))
- Substitute given values to solve for y: y = 1.7 + 2.5*tan(arctan(0.55/2.5)) = 2.15 m
- Answer to a: The bottom edge of the mirror is 2.15 m above the floor.
- Use similar triangles to determine the new horizontal distance, say x', that the person sees when he doubles his distance from the mirror. Since the triangles are similar, we have (y-h)/d = (y-h')/(2d), where h' is the new height of the person's eyes. Rearranging gives h' = y - (d/2)*[(y-h)/d], which simplifies to h' = (2y-h)/2.
- Substitute the value of y from step 3: h' = (2*2.15 - 1.7)/2 = 1.825 m.
- Use the law of reflection to determine the corresponding horizontal distance x'. We have x'/2.5 = (h - y)/y, which simplifies to x' = 2.5*(h-y)/y.
- Substitute the values of h and y from steps 1 and 3, respectively: x' = 2.5*(1.7 - 2.15)/2.15 = -0.424.
- Answer to b: The person will see a horizontal distance of 0.424 m to the left of the mirror.