A person whose eyes are H = 1.62 m above the floor stands L =2.24m in front of a vertical plane mirror whose bottom edge is 43 cm above the floor. What is the horizontal distance x to the base of the wall supporting the mirror of the nearest point on the floor that can be seen reflected in the mirror?

Draw a line from the person's eyes to the base of the mirror. Next, imagine that to be a ray traveling to the person's eyes, and draw the ray that reflects specularly on the mirror to form that ray. See where it intersects the floor. That is where the nearest point can be seen.

The rays I have described are inclined arctan 0.81/2.24 = 19.9 degrees to horizontal.

I get the distance to be 0.43m*cot 19.9 = 1.19 m

See what you get. You need to draw the figure.

To find the horizontal distance (x) to the base of the wall supporting the mirror of the nearest point on the floor that can be seen reflected in the mirror, we can use similar triangles.

First, let's label the important points:

- Point A: Base of the wall supporting the mirror
- Point B: Bottom edge of the mirror
- Point C: Person's eyes
- Point D: Nearest point on the floor

We have the following information:

- Height of the person's eyes above the floor (AC): H = 1.62 m
- Distance between the person and the mirror (from eyes to mirror): L = 2.24 m
- Height of the bottom edge of the mirror above the floor (CB): 43 cm = 0.43 m

Using similar triangles, we can set up the following ratios:

1) AC / AB = CB / DB
2) AB = L
3) CB = 0.43 m

We want to find DB, which is the horizontal distance to the base of the wall. Rearranging equation (1) to solve for DB:

DB = CB * AB / AC

Substituting the known values:

DB = (0.43 m) * (2.24 m) / (1.62 m)

Now we can calculate the value of DB:

DB ≈ 0.592469 m

Therefore, the horizontal distance (x) to the base of the wall supporting the mirror of the nearest point on the floor that can be seen reflected in the mirror is approximately 0.592469 meters.

To find the horizontal distance (x) to the base of the wall supporting the mirror, we can use the concept of similar triangles.

First, let's label the relevant points in the problem:

- The person's eye position is marked as E
- The vertical plane mirror is marked as M
- The point where the bottom edge of the mirror meets the floor is marked as A
- The base of the wall supporting the mirror is marked as B
- The nearest point on the floor that can be seen reflected in the mirror is marked as C

We can form two right-angled triangles, one with vertices E, B, and C, and the other with vertices E, M, and A.

Now, let's break down the information given:

- H = 1.62 m is the vertical height of the person's eyes above the floor (distance from E to A).
- L = 2.24 m is the distance from the person to the mirror (distance from E to M).
- The bottom edge of the mirror is 43 cm above the floor.

To start solving, let's convert the 43 cm to meters:

43 cm = 43/100 m = 0.43 m

Now, let's calculate the horizontal distance x:

Using the concept of similar triangles, we have the following ratio:

x / L = (H - 0.43) / H

We can rearrange this equation to solve for x:

x = L * ((H - 0.43) / H)

Substituting the given values:

x = 2.24 * ((1.62 - 0.43) / 1.62)

x = 2.24 * (1.19 / 1.62)

x ≈ 1.64 meters

Therefore, the horizontal distance (x) to the base of the wall supporting the mirror, from the nearest point on the floor that can be seen reflected in the mirror, is approximately 1.64 meters.