A shaving/makeup mirror is designed to magnify your face by a factor of 1.33 when your face is placed 20.0 cm in front of it. (a) What type of mirror is it? (b) Describe the type of image that it makes of your face. (c) Calculate the required radius of curvature for the mirror.

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To determine the type of mirror, we need to consider the magnification factor and the distance between the object and the mirror.

(a) The magnification factor of 1.33 implies that the mirror is a concave mirror. Concave mirrors are curved inward and can magnify objects when they are placed within the focal length of the mirror.

(b) When your face is placed 20.0 cm in front of the mirror, the concave mirror will create a magnified image of your face. Since the magnification factor is greater than 1, the image will appear larger than the actual size of your face.

(c) To calculate the required radius of curvature for the mirror, we can use the mirror formula:

1/f = 1/do + 1/di

Where:
f is the focal length of the mirror,
do is the distance between the object and the mirror, and
di is the distance between the image and the mirror.

Since the object distance (do) is given as 20.0 cm and the magnification factor (m) is 1.33, we can find the image distance (di) using the formula for magnification:

m = -di/do

Rearranging the formula, we can solve for di:

di = -m * do

Substituting the given values:

di = -1.33 * 20.0 cm
di = -26.6 cm

Now, we can substitute the values of do and di into the mirror formula to find the focal length (f):

1/f = 1/do + 1/di

1/f = 1/20.0 cm + 1/(-26.6 cm)

Simplifying the equation gives:

1/f = 0.05 cm^(-1) - 0.037 cm^(-1)

1/f = 0.013 cm^(-1)

Taking the reciprocal of both sides:

f = 1/0.013 cm^(-1)

f ≈ 76.9 cm

Therefore, the required radius of curvature for the mirror is approximately 76.9 cm.