# A concave makeup mirror is designed so that a person 25 cm in front of it sees an upright image...

## Question:

A concave makeup mirror is designed so that a person 25 cm in front of it sees an upright image magnified by a factor of two.

What is the radius of curvature of the mirror?

## Concave Mirror:

Concave mirror forms an image that is inverted and real if the object is placed at a distance greater than focal length. In order to form an upright image, the object must be placed at a distance less than focal point.

Solving for the focal length, we used:

{eq}\frac{1}{f} = \frac{1}{s} + \frac{1}{s'} {/eq}

And radius of curvature of a mirror is equal to {eq}R = 2f {/eq}

Given:

s = 25 cm

Solution:

Magnification is given by the formula:

{eq}m = \frac{-s'}{s} \\ 2 = \frac{-s'}{s} {/eq}

Since the image is said to be upright, therefore the magnification is +2. And the distance of the image formed is negative since the incoming light and outgoing light are not on the same side.

{eq}s' = -ms \\ s' = -(2)(25 \ cm) \\ s' = -50 \ cm {/eq}

Radius of Curvature is twice the focal length where focal length is equal to:

{eq}\frac{1}{f} = \frac{1}{s} + \frac{1}{s'} \\ \frac{1}{f} = \frac{1}{25 \ cm} + \frac{1}{-50 \ cm'} \\ f = 50 \ cm {/eq}

Therefore, Radius of Curvature of concave mirror is:

{eq}R = 2f \\ R = 100 \ cm {/eq}