A convex mirror has a focal length of −15 cm. What is the object distance if the image distance is −8.5 cm? Answer in units of cm

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The focal length f, the distance of the object Do and distance of the image Di for a mirror is related by 1/Do + 1/Di = 1/f. The sign of f is negative in the case of convex mirrors, Do is positive as the object is placed in front of...

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The focal length f, the distance of the object Do and distance of the image Di for a mirror is related by 1/Do + 1/Di = 1/f. The sign of f is negative in the case of convex mirrors, Do is positive as the object is placed in front of the mirror, Di can be positive or negative depending on whether the image is in front or behind the mirror.

Substituting the values given, f = -15 and Di = -8.5,

1/Do - 1/8.5 = -1/15

=> 1/Do = 13/255

=> Do = 255/13

The object distance for which the image distance is 8.5 cm behind the mirror is 255/13 cm. The object is placed in front of the mirror.

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For focal length of a convex mirror, we use the following formula where f is the focal length, do is the distance of the object and di is the distance of the image.

1/f = 1/do + 1/di

Since the focal length and image distance are both in cm, it is not necessary to do any conversions prior to doing the calcultion.  As a result, the object distance will also be in units of cm.

First substitute into the known values and then solve for object distance.  Depending on your calculator skills, you may be able to solve this in fewer steps but each individual step is shwon below.

1/(-15) = 1/do + 1/(-8.5)

-0.0667 = 1/do -0.118

1/do = 0.0509

do = 19.6 cm as the object distance.

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