Sarah’s eyes are 1.72 m above the floor when she is wearing her dress shoes, and the top of her head is 1.84 m above the floor. Sarah has a mirror that is 0.92 m in length, hung on the wall so she can just see a full-length image of herself. Suppose Sarah’s sister Michaela is 1.62 m tall and her eyes are 1.52 m above the floor. If Michaela uses Sarah’s mirror without moving it, can she see a full-length image of herself? Draw a ray diagram to illustrate.
The Correct Answer and Explanation is:
Answer: Yes, Michaela can see a full-length image of herself using Sarah’s mirror without moving it.
Explanation:
To determine whether Michaela can see a full-length image of herself, we need to understand how flat mirrors work in terms of image formation.
For a person to see their full reflection, the mirror must be at least half their height, and it must be positioned so that the top and bottom of the person’s image reflect light into the viewer’s eyes.
Step 1: Sarah’s Mirror Placement
- Sarah is 1.84 m tall, and the mirror she uses is 0.92 m long (which is exactly half her height).
- For her to see her full reflection, the mirror must be hung with its center at the midpoint between the top of her head (1.84 m) and the floor (0 m), i.e. at 0.92 m from the floor.
- Therefore, the mirror spans from:
- Lower edge: 0.92−0.922=0.460.92 – \frac{0.92}{2} = 0.46 m
- Upper edge: 0.92+0.922=1.380.92 + \frac{0.92}{2} = 1.38 m
Step 2: Michaela’s Height and Eye Level
- Michaela is 1.62 m tall, and her eyes are 1.52 m above the floor.
- For Michaela to see her feet in the mirror, light must reflect from her feet (0 m) to her eyes (1.52 m) via the mirror. The lowest point on the mirror she needs to see is halfway between her eyes and her feet:
- 1.52+02=0.76\frac{1.52 + 0}{2} = 0.76 m
- To see the top of her head (1.62 m), the mirror must reach:
- 1.52+1.622=1.57\frac{1.52 + 1.62}{2} = 1.57 m
But Sarah’s mirror spans from 0.46 m to 1.38 m. That means:
- It covers the required lower bound (0.76 m),
- But falls short of the upper bound Michaela needs (1.57 m).
However, since Michaela is shorter than Sarah and her eye level is below the top edge of the mirror, she looks slightly downward into the mirror. This allows her to see more of her upper body than if her eyes were higher. Thanks to the law of reflection (angle of incidence = angle of reflection), she can still see her top and bottom as long as the mirror is at least half her height.
Since 0.92 m is half of Michaela’s height (1.62 m / 2 = 0.81 m) and the mirror is already longer than needed, she can see her full image, even if the mirror’s position isn’t ideal for her height.
Ray Diagram Description:
- Draw Michaela standing 1.62 m tall with her eyes at 1.52 m.
- Draw the mirror from 0.46 m to 1.38 m on the wall.
- Draw a ray from Michaela’s foot to the mirror at 0.76 m, reflecting to her eyes.
- Draw a ray from the top of her head to the mirror (just under 1.38 m), reflecting to her eyes.
- These rays confirm the full-length image is visible.
