A dog sitting 2.2m from a vertical mirror thinks its image is another dog.

A)How far is the image from the dog?

B)To investigate, the dog walks directly toward the mirror at 0.28m/s . At what rate does the dog approach its image?

The image is 4.4 m from the dog.

b. rate relative to the dog, or surroundings? Relative to the dog, .56m/s, relative to wall, .28m/s

A) 4.4m

B) .56 m/s

To answer these questions, we can use the principles of reflection and the geometry of the situation. Let's start with the first question:

A) How far is the image from the dog?

To determine the distance of the image from the dog, we need to understand that when light reflects off a mirror, the angle of incidence is equal to the angle of reflection. In this case, the dog sees the reflected image as another dog, which means the angle of incidence (angle between the dog's line of sight and the mirror) is equal to the angle of reflection (angle between the mirror and the line of sight of the reflected image).

Since the dog thinks its image is at the same distance from the mirror, we can use basic geometry to find the distance of the image.

If the distance between the dog and the mirror is 2.2m, we can consider this as the hypotenuse of a right triangle. The distance of the image will be the adjacent side of this triangle.

Let's call the distance of the image "x" (in meters). Using trigonometry, we can say:

cos(angle of incidence) = adjacent/hypotenuse
cos(angle of incidence) = x/2.2

Now, we need to find the angle of incidence. Since the dog sees its image as another dog, it must be looking at the image at the same angle it would look at another dog.

Given that the dog is looking straight ahead, the angle of incidence can be determined as the angle between the dog's line of sight and a line perpendicular to the mirror.

This angle is formed by the perpendicular line, the dog's line of sight, and the reflected line. It is equal to twice the angle between the dog's line of sight and the mirror. We can call this angle "theta".

To find theta, we can use the fact that a straight line and its reflection form a 180-degree angle. Therefore, we have:

2 * theta + 180 degrees = 180 degrees

Simplifying the equation, we find:

2 * theta = 0 degrees

This tells us that theta is zero degrees, which means the angle of incidence is 0 degrees as well. Cos(0 degrees) is equal to 1, so we have:

1 = x/2.2

Solving for x, we get:

x = 2.2 meters

Therefore, the image is also 2.2 meters from the dog.

B) At what rate does the dog approach its image?

To find the rate at which the dog approaches its image, we need to consider the speed at which the dog is moving and the change in distance between the dog and the image over time.

The dog is walking toward the mirror at a constant speed of 0.28 meters per second. This represents the rate of change of the dog's distance from the mirror.

Since the image is the same distance from the mirror as the dog, it will also move towards the mirror at the same rate.

Therefore, the rate at which the dog approaches its image is 0.28 meters per second.