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# When a man's face is in front of a concave mirror of radius 100 cm, the lateral magnification of the image is +1.5. What is the image distance?

Solution

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Given for a concave mirror:

$\begin{gather*} \text{Radius of curvature} \ R=100\ \mathrm{cm}\\ \text{Lateral magnification of image} \ M=+1.5 \end{gather*}$

To determine the image distance.

Using Spherical mirror equation and Magnification factor equation [23.3] & [23.4]

$\begin{gather} M=-\frac{d_i}{d_o}\\ \frac{1}{d_i}+\frac{1}{d_o}=\frac{1}{f}=\frac{2}{R} \end{gather}$

$\begin{gather*} \text{Using equation }(1)\\ M=\frac{d_i}{d_o}=+1.5\\ \implies -\frac{d_i}{1.5}=d_o\\ \implies \frac{1}{d_o}=-\frac{1.5}{d_i}\\ \text{Putting this in equation (2) } \\ \implies \frac{1}{d_i}-\frac{1.5}{d_i}=\frac{2}{100}\\ \implies -\frac{0.5}{d_i}=\frac{1}{50}\\ \implies d_i=-50\times 0.5\\ \implies \boxed{d_i=-25\ \mathrm{cm}} \end{gather*}$

Thus the distance of image is $25\ \mathrm{cm}$ behind the mirror(negative sign).

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