An object is 24 cm away from a concave mirror and its image is 16 cm from the mirror. Find the focal length and radius of curvature of the mirror, and the magnification of the image.

Hint: Use the lens formula to determine the focal length of the mirror. The image formed by the concave mirror is on the same side of the lens as the object. The radius of curvature of the mirror or lens is twice the focal length of the mirror.

Formula used:
\[\dfrac{1}{f} = \dfrac{1}{v} + \dfrac{1}{u}\]
Here, f is the focal length, v is the object distance and u is the image distance.

Complete step by step answer:For a concave mirror, the focal length is negative. Also, the object and image are on the same side of the mirror. therefore, the distance of both object and image is negative.

Use lens formula to determine the focal length of the concave mirror as follows,
\[\dfrac{1}{f} = \dfrac{1}{v} + \dfrac{1}{u}\]

Here, f is the focal length, v is the object distance and u is the image distance.

Substitute \[ - 16\,cm\] for v and \[ - 24\,cm\] for u in the above equation.
\[\dfrac{1}{f} = \dfrac{1}{{ - 16}} + \dfrac{1}{{ - 24}}\]
\[ \Rightarrow \dfrac{1}{f} = \dfrac{{ - \left( {24 + 16} \right)}}{{384}}\]
\[ \Rightarrow \dfrac{1}{f} = - 0.104\]
\[\therefore f = - 9.6\,cm\]

Therefore, the focal length of the given concave mirror is \[ - 9.6\,cm\].

 The relation between the radius of curvature and focal length is,
\[R = 2f\]

Substitute \[ - 9.6\,cm\] for f in the above equation.
 \[R = 2\left( { - 9.6\,cm} \right)\]
\[\therefore R = - 19.2\,cm\]

Therefore, the radius of the curvature of the given mirror is \[ - 19.2\,cm\].

The formula for the magnification of the mirror is,
\[m = - \dfrac{v}{u}\]

Substitute \[ - 16\,cm\] for v and \[ - 24\,cm\] for u in the above equation.
\[m = - \dfrac{{ - 16\,cm}}{{ - 24\,cm}}\]
\[m = - \dfrac{2}{3}\]

Therefore, the magnification of the given concave mirror is \[ - \dfrac{2}{3}\].

Note:Always choose the positive and negative scale for the object distance and image distance. For simplicity, take the distance towards the left from the mirror as negative and the distance towards the right of the lens as positive.