Question

Eve, the 176 cm tall model is looking at herself in a mirror hanging on the wall. The mirror is vertical and its height is 75 cm. Eve stands straight, her eyes are at a height of 168 cm from the floor. How much length (in cm) of her image can she view if she can just see her shoes?

Answer 1

150

Answer 2

Let $H$ be the total height of Eve, $H = 176$ cm.
Let $h_e$ be the height of Eve's eyes from the floor, $h_e = 168$ cm.
Let $H_m$ be the height of the mirror, $H_m = 75$ cm.

To see any point on her body, say at height $y_{body}$ from the floor, a ray of light from this point must reflect off the mirror at a point at height $y_P$ and reach her eyes at height $h_e$. Using the law of reflection or considering the image formed behind the mirror, the height of the reflection point $y_P$ is related to the height of the body part $y_{body}$ and the eye height $h_e$ by the formula: $y_P = \frac{y_{body} + h_e}{2}$

Conversely, if the reflection occurs at a point on the mirror at height $y_P$, the height of the corresponding point on the body whose image is seen is: $y_{body} = 2y_P - h_e$

The problem states that Eve can just see her shoes. Her shoes are at height $y_{shoes} = 0$ cm (assuming the floor is the reference level).
For her to see her shoes, the ray from her shoes must reflect off the bottom edge of the mirror and reach her eyes.
Let $y_{bottom}$ be the height of the bottom edge of the mirror from the floor.
The height of the reflection point for seeing the shoes is $y_P = \frac{y_{shoes} + h_e}{2} = \frac{0 + 168}{2} = 84$ cm.
Since she can just see her shoes, the bottom edge of the mirror must be at this height.
So, $y_{bottom} = 84$ cm.

The height of the mirror is given as $H_m = 75$ cm.
The top edge of the mirror is at a height $y_{top} = y_{bottom} + H_m = 84 + 75 = 159$ cm from the floor.

The part of her image she can view is determined by the range of heights on the mirror from which rays can reflect and reach her eyes. This range is the height of the mirror itself, from $y_{bottom}$ to $y_{top}$.

The lowest point of her body whose image she can see corresponds to the reflection from the bottom edge of the mirror $(y_P = y_{bottom} = 84$ cm). The height of this point on her body is: $y_{low} = 2 \times y_{bottom} - h_e = 2 \times 84 - 168 = 168 - 168 = 0$ cm.
This confirms that she can see her shoes (at height 0 cm).

The highest point of her body whose image she can see corresponds to the reflection from the top edge of the mirror $(y_P = y_{top} = 159$ cm). The height of this point on her body is: $y_{high} = 2 \times y_{top} - h_e = 2 \times 159 - 168 = 318 - 168 = 150$ cm.

So, Eve can view the image of her body from a height of 0 cm (her shoes) up to a height of 150 cm.
The length of the image she can view is the difference between the heights of the highest and lowest visible points on her body.
Length of image viewed = $y_{high} - y_{low} = 150 - 0 = 150$ cm.

The length of the image seen in a plane mirror is equal to the length of the object segment being viewed. Thus, the length of her image she can view is 150 cm.