10. An object is 30 cm from a concave mirror that has a focal length of 12 cm . Determine the image location, magnification, real or virtual, and upright or inverted for the object. a. Using the equation:

To find the image location, we use the mirror formula: \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \). Here, \( f = -12 \) cm (focal length of a concave mirror is negative), and \( d_o = -30 \) cm (object distance is also negative in mirror conventions). Rearranging gives \( \frac{1}{d_i} = \frac{1}{f} - \frac{1}{d_o} \). Plugging in the values results in \( d_i = 10 \) cm, which means the image is located 10 cm in front of the mirror. To find the magnification (\( m \)), we use the equation \( m = -\frac{d_i}{d_o} \). Using the image and object distances calculated, we find \( m = -\frac{10}{30} = -\frac{1}{3} \). The negative sign indicates that the image is inverted, and since magnification is less than one, the image is smaller than the object. Thus, the image is real, inverted, and located 10 cm from the mirror. For more practical applications, concave mirrors are often used in shaving mirrors or makeup mirrors where a magnified image is beneficial. The reflection can make facial details clearer, offering a practical and handy use of concave mirrors in everyday life! In the realm of optics, many people confuse the rules of sign convention, especially with distance measurements. A common mistake is using the wrong sign for the focal length of concave mirrors or the object distance. Remember, for concave mirrors, the focal length is negative, and the object distance is considered positive only when the object is located in front of the mirror (real object). Keeping these sign conventions straight will save you from potential errors in future calculations!