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How far from the mirror surface is the image in the following case: Convex mirrors are used in stores to provide a wide angle of surveillance for a reasonable mirror size.  A mirror with a radius of...

How far from the mirror surface is the image in the following case:

Convex mirrors are used in stores to provide a wide angle of surveillance for a reasonable mirror size.  A mirror with a radius of curvature 1 m allows a clerk 3 m away from the mirror to survey the entire store.  If a customer is 6 m from the mirror, how far, in meters, from the mirror surface is his image?

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The focal length of a convex mirror is half the length of its radius of curvature. Here the radius of curvature is given as 1 m. This gives the focal length as 0.5 m.

The relation between the focal length f, the distance of the object from the mirror Do and the distance of the image Di from the mirror for a convex mirror is (1/f) = (1/Do) + (1/Di).

We are given that the customer is 6 m away from the mirror, Do = 6.

Substituting the values in the equation given earlier:

1/-0.5 = 1/6 + 1/Di

=> 1/Di = 1/-0.5 - 1/6

=> -2 - 1/6

=> -13/6

=> -2.167 m

The image of the customer is 2.167 m away from the surface of the mirror.

As the distance of the image is negative we know that it is a virtual image. A convex mirror always forms a virtual image that is smaller than the object.

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