If you look at yourself in a shiny Christmas tree ball with a diameter of 10.5 cm when your face is 35.0 cm away from it, (a) where is your image? (b) Is it real or virtual? (c) Is it upright or inverted?

To determine the answers to your questions, we can use the thin lens equation. The equation is as follows:

1/f = 1/di + 1/do

Where:
- f is the focal length of the lens
- di is the distance of the image from the lens
- do is the distance of the object from the lens

Given that the shiny Christmas tree ball has a diameter of 10.5 cm, we can assume it acts as a convex mirror with a focal length of half its radius, which is 5.25 cm.

(a) Determining the location of your image:
Using the thin lens equation, we can substitute the values and solve for di. Assuming your face is the object and it is located 35.0 cm away,

1/5.25 = 1/di + 1/35.0

Simplifying the equation, we have:

1/di = 1/5.25 - 1/35.0

Calculating the value, we get:

1/di = 0.1905

Taking the reciprocal on both sides, we find that:

di = 5.25 cm

Therefore, the image of your face is formed at a distance of 5.25 cm from the shiny Christmas tree ball.

(b) Determining if the image is real or virtual:
To determine if the image is real or virtual, we look at the sign of the image distance. Given that the image distance (di) is positive (5.25 cm), the image is formed on the opposite side of the object. Thus, the image is virtual.

(c) Determining if the image is upright or inverted:
The image formed by a convex mirror is always virtual, and virtual images are always upright. Therefore, the image of your face in the shiny Christmas tree ball will be upright.

To answer these questions, we can use the concepts of reflection and the properties of concave mirrors. Here's how you can determine the answers:

(a) To find where your image is formed, you need to identify the focal point and center of curvature of the mirror. For a concave mirror, the center of curvature (C) is located twice the focal length (F) away from the mirror's surface. The focal length (F) can be determined using the mirror's radius of curvature (R) using the equation F = R/2.

Given that the diameter of the Christmas tree ball (which can be considered a spherical mirror) is 10.5 cm, the radius of curvature can be calculated as half the diameter, which is 5.25 cm. Hence, R = 5.25 cm.

Using the formula F = R/2, we find that the focal length (F) is 2.625 cm.

Now, to find the image distance (di), we can use the mirror equation:

1/f = 1/do + 1/di

where do is the object distance (the distance between your face and the mirror).

Given that do = 35.0 cm and f = 2.625 cm, we can rearrange the equation to solve for di:

1/di = 1/f - 1/do

Substituting the values, we get:

1/di = 1/2.625 - 1/35.0

Calculating this expression, we find that 1/di = 0.381. Taking the reciprocal of both sides, we get di = 2.62 cm.

Therefore, your image is formed at a distance of 2.62 cm from the mirror.

(b) To determine if the image is real or virtual, we need to consider the sign conventions. In this case, since the object is located beyond the focal point of the mirror (do > f), the image will be real.

(c) To determine if the image is upright or inverted, we need to analyze the characteristics of the image formed by a concave mirror. When the object is placed beyond the focal point of a concave mirror, like in this case, the image formed is real and inverted.

Therefore:
(a) Your image is formed at a distance of 2.62 cm from the mirror.
(b) The image is real.
(c) The image is inverted.