What will be the image distance when he reverses the mirror ?

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The discussion revolves around calculating the image distance when a man reverses a double-sided spherical mirror from convex to concave while maintaining a distance of 47.4 cm from his face. The initial calculations indicate that the focal length for the convex side is positive, while for the concave side, it should be negative. The user struggles with sign conventions and the resulting image distance, which is calculated to be approximately -8.53 cm for the concave mirror. Suggestions include ensuring the correct application of sign conventions for object and focal distances, as well as verifying calculations for accuracy. The overall goal is to achieve a correct understanding of the mirror formula and magnification in both configurations.
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Homework Statement



A man holds a double-sided spherical mirror so that he is looking directly into its convex surface, 47.4 cm from his face. The magnification of the image of his face is +0.18. What will be the image distance when he reverses the mirror (looking into its concave surface), maintaining the same distance between the mirror and his face? Be sure to include the algebraic sign (+ or −) with your answer.



Homework Equations



mirror formula> 1/f = 1/v + 1/u
f = vu/(v+u)
v> image, u>object, f> focal
-----------------

m = magnification = - v/u

1/f = 1/v + 1/u
v = fu/(u-f)


The Attempt at a Solution



mirror formula> 1/f = 1/v + 1/u
f = vu/(v+u)
v> image, u>object, f> focal
-----------------
convex> u = - 47.4,
m = magnification = - v/u = - v/(-47.4) = +0.18
v = 8.532
f = 8.532*47.4/8.532-47.4 = 7.23
-----------------------------
it has to be assumed that radius of curvature (R = 2f) is same
concave>
f = - 7.23
u = - 47.4 (same)
1/f = 1/v + 1/u
v = fu/(u-f)
v = concave = (-7.23)(-47.4)/[- 47.4+ 7.23]
v = concave = - 8.53129cm
m = - 8.531 /47.4 = - 0.17998


I am getting the wrong answer with the wrong sign somehow, but I don't know where I went wrong. Any help would be greatly appreciated. Thank you!
 
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By convention, object distance should always be positive for a real object. The focal length should be negative for a convex mirror and positive for a concave one. If you flip all your signs around, you'll get the right answer.
 
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