If you look at yourself in a shiny Christmas tree ball with a diameter of 9.0 cm when your face is 38.0 cm away from it, where is your image? (Your answer should be positive if the image is in front of the ball's surface, and negative if the image is behind it.)
To determine the location of the image, we can use the concept of reflection and the mirror equation, which relates the object distance (distance of the face from the ball) to the image distance (location of the image) and the focal length of the spherical Christmas tree ball.
The mirror equation is given by:
1/f = 1/d_o + 1/d_i
Where:
f = focal length of the mirror (Christmas tree ball)
d_o = object distance (distance of the face from the ball)
d_i = image distance (location of the image)
Since the Christmas tree ball is a convex mirror, its focal length is negative. In this case, the focal length can be taken as -9.0 cm.
Substituting the given values:
1/(-9.0 cm) = 1/(38.0 cm) + 1/d_i
Now, we can solve for d_i:
1/d_i = -1/9.0 cm + 1/38.0 cm
1/d_i = (-4/36) + (1/38)
1/d_i = (-2/18) + (1/38)
1/d_i = (-38/342) + (9/342)
1/d_i = (-29/342)
d_i = -342/29 cm
d_i ≈ -11.8 cm
The image is located approximately 11.8 cm behind the surface of the Christmas tree ball. Since the image is behind the ball, the answer is negative.