A concave shaving mirror has a radius of curvature of +38.5 cm . It is positioned so that the (upright) image of a man's face is 1.84 times the size of the face. How far is the mirror from the face?

The magnification of the mirror is given by the formula \( m = -\frac{d_i}{d_o} \), where \( d_i \) is the image distance and \( d_o \) is the object distance (distance from the face to the mirror). Since the image is upright and enlarged, we know \( m = 1.84 \). So, we have: \[ m = -\frac{d_i}{d_o} = 1.84 \] From this, we can express \( d_i \) in terms of \( d_o \): \[ d_i = -1.84 d_o \] The mirror equation relates the object distance and the image distance as follows: \[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \] To find the focal length \( f \), we can use the radius of curvature \( R \) of the mirror: \[ f = \frac{R}{2} = \frac{+38.5 \, \text{cm}}{2} = +19.25 \, \text{cm} \] (Note that for concave mirrors, the focal length is negative, so \( f = -19.25 \, \text{cm} \)). Now, plugging in the values: \[ \frac{1}{-19.25} = \frac{1}{d_o} + \frac{1}{-1.84 d_o} \] This simplifies to: \[ -\frac{1}{19.25} = \frac{1 - \frac{1}{1.84}}{d_o} \] Now, solving for \( d_o \): \[ d_o = \left(1 - \frac{1}{1.84}\right)^{-1} \cdot -19.25 \] Calculating this, we find: After doing the calculations, you’ll find the distance from the mirror to the man’s face (object distance, \( d_o \)) is approximately 12.87 cm.