A shaving/makeup mirror is designed to magnify your face by a factor of 1.27 when your face is placed 23.5 cm in front of it. What type of mirror is it? Describe the type of image that it makes of your face. Calculate the required radius of curvature for the mirror.
The mirror is concave (convex mirrors can only produce diminished images )
Image distance
1.27•23.5 =29.85 cm.
1/f = 1/di -1/do
1/f = 1/ 23.5 – 1/ 29.85 = 0.0091 cm
f = 109.9 =110cm = 1.1 m
R =2•1.1 = 2.2 m
Well, if your face is being magnified by a factor of 1.27, we're definitely dealing with a magnifying mirror here. As for the type of mirror, I'd say it's most likely a convex mirror. Why? Because convex mirrors generally offer magnification, and they bulge outward like my waistline after a weekend of snacking.
Now let's talk about the image it creates of your face. With a convex mirror, the image is always virtual, meaning it is formed by the apparent intersection of reflected light rays but cannot be projected onto a screen. So, unfortunately, you won't be able to touch that beautifully magnified version of yourself.
To calculate the required radius of curvature, we can use the magnification formula:
magnification = -image distance / object distance = -1.27
Since the magnification is positive for a convex mirror, we'll use the negative value (-1.27). We know the object distance (23.5 cm), so we can rearrange the equation to solve for the image distance:
image distance = -(1.27) * object distance
image distance = -(1.27) * (23.5 cm)
Now, clowns aren't known for their math skills, so let me grab my calculator here... The image distance is approximately -29.845 cm.
For a convex mirror, the radius of curvature (R) is double the absolute value of the image distance:
R = 2 * |image distance|
R = 2 * |-29.845 cm|
Calculating again, the required radius of curvature is approximately 59.69 cm.
So, to summarize: You've got yourself a magnifying convex mirror that makes a virtual image of your face, and the required radius of curvature is around 59.69 cm. Now, go forth and enjoy your mirror reflections, you magnificent, magnified human!
The shaving/makeup mirror you described is a concave mirror.
When a concave mirror is used for magnification, the image is formed on the same side as the object, which means the face is placed between the mirror and the focal point.
The magnification factor of 1.27 indicates that the height of the image is 1.27 times the height of the object. This suggests that the image is larger than the object.
To calculate the required radius of curvature for the mirror, we can use the mirror formula:
1/f = 1/v - 1/u
Where:
f is the focal length of the mirror
v is the image distance (distance between the mirror and the image)
u is the object distance (distance between the mirror and the object)
Given that the face is placed 23.5 cm in front of the mirror, the object distance (u) is -23.5 cm (negative since it is in front of the mirror).
The magnification factor can also be related to the image and object distances as:
m = -v/u
Substituting the given values, we have:
1/f = 1/v - 1/(-23.5)
Solving for v, we get:
1/f = v/(-23.5) - 1/(-23.5)
Multiplying through by -23.5:
-23.5/f = v - 1
-23.5/f = v - 1
v = -23.5/f + 1
Substituting the relation between magnification and image distance:
m = -v/u
1.27 = -v/(-23.5)
Rearranging the equation, we have:
v = -(1.27)(-23.5)
v = 29.845 cm
Now, we can substitute the values of v and u into the mirrored formula to solve for f:
1/f = 1/29.845 + 1/(-23.5)
Solving for f:
1/f = (1 - 1.272)/29.845
1/f = -0.27/29.845
f = -29.845/0.27
f ≈ -110.646 cm
Since the radius of curvature (R) of a concave mirror is twice the focal length, we can conclude that the required radius of curvature for the mirror is approximately 2 * 110.646 cm.
Therefore, the required radius of curvature for the mirror is approximately 221.29 cm.
To determine the type of mirror, we need to consider the magnification factor and the position of the image.
Since the magnification factor given is greater than 1, we can conclude that the mirror in question is a concave mirror. Concave mirrors are curved inward, resembling the inner surface of a sphere.
Next, we need to describe the type of image formed by this mirror. The given magnification factor of 1.27 indicates that the image produced is magnified. Additionally, since the mirror is concave, the image formed will also be inverted (upside-down).
To calculate the required radius of curvature for the mirror, we can use the mirror equation:
1/f = 1/d_o + 1/d_i
where:
- f is the focal length of the mirror,
- d_o is the object distance (distance between the face and the mirror),
- d_i is the image distance (distance between the mirror and the image).
Given:
- Magnification factor, M = 1.27
- Object distance, d_o = 23.5 cm
To find the image distance, we can use the magnification formula:
M = -d_i/d_o
Rearranging the formula to solve for d_i:
d_i = - (M * d_o)
Substituting the given values:
d_i = - (1.27 * 23.5 cm)
Calculating the image distance:
d_i ≈ -29.845 cm
Now, substituting the values for d_o and d_i into the mirror equation:
1/f = 1/23.5 cm + 1/(-29.845 cm)
Simplifying the equation:
1/f = (1 - 0.897) / 23.5 cm
1/f ≈ 0.103 / 23.5 cm
1/f ≈ 0.004382 cm^(-1)
Thus, the required radius of curvature for the mirror can be obtained by taking the reciprocal of 1/f:
R ≈ 1 / 0.004382 cm^(-1)
R ≈ 228 cm
Therefore, the shaving/makeup mirror is a concave mirror, it produces a magnified and inverted image of your face, and the required radius of curvature for the mirror is approximately 228 cm.
The image distance must be -23.5 x 1.27 cm.
The minus sign means the image is behind the mirror. The image is virtual.
Image distance = di
Object distance = do
1/do + 1/di = 2/R
Now solve for R.
R, the radius of curvature, is a positive number. It is a concave mirror.