A spherical concave mirror has a radius of curvature of 6.0 cm.At what distance from the mirror should a 6.0-cm object be placed to obtain an image that is 48 cm tall?
A)1.3 cm
B)3.6 cm
C)4.2 cm
D)5.3 cm
E)6.8 cm

Question

Quizplus · Accepted Answer

The magnification (m) of a mirror is given by the formula $m = \frac{-v}{u} = \frac{h_i}{h_o}$, where $v$ is the image distance, $u$ is the object distance, $h_i$ is the image height, and $h_o$ is the object height. Given that the image height is 48 cm and the object height is 6 cm, the magnification is $m = \frac{48}{6} = 8$. The negative sign indicates that the image is inverted, which is typical for concave mirrors when the object is placed beyond the focal point.The mirror equation is $\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$, where $f$ is the focal length. The radius of curvature $R = 6.0$ cm, so the focal length $f = \frac{R}{2} = 3.0$ cm.Substituting the magnification into the mirror equation gives us two equations:1. $m = \frac{-v}{u} = 8$2. $\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$From 1, we get $v = -8u$. Substituting $f = 3.0$ cm and $v = -8u$ into the mirror equation, we get:$\frac{1}{3} = \frac{1}{-8u} + \frac{1}{u}$Solving for $u$, we find that $u$ is approximately 3.6 cm, which means the object should be placed 3.6 cm from the mirror to obtain the described image.

A Spherical Concave Mirror Has a Radius of Curvature of 6.0