9. An object is 15 cm from a concave mirror that has a focal length of 10 cm . Determine the image location, magnification, real or virtual, and upright or inverted for the object. a. Using the equation:

To solve this problem, we will use the mirror formula and the magnification formula for concave mirrors. ### Known Conditions: - Object distance (\(d_o\)) = -15 cm (the object distance is taken as negative in mirror conventions) - Focal length (\(f\)) = -10 cm (the focal length is negative for concave mirrors) ### Step 1: Use the Mirror Formula The mirror formula is given by: \[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \] Where: - \(d_i\) = image distance (what we want to find) ### Step 2: Rearranging the Mirror Formula We can rearrange the formula to solve for \(d_i\): \[ \frac{1}{d_i} = \frac{1}{f} - \frac{1}{d_o} \] ### Step 3: Substitute the Known Values Now, substituting the known values into the equation: \[ \frac{1}{d_i} = \frac{1}{-10} - \frac{1}{-15} \] ### Step 4: Calculate \(d_i\) Let's calculate \(d_i\) using the above expression. \[ \frac{1}{d_i} = -\frac{1}{10} + \frac{1}{15} \] To perform this calculation, we will find a common denominator and simplify. Calculating \(d_i\): \[ \frac{1}{d_i} = -\frac{3}{30} + \frac{2}{30} = -\frac{1}{30} \] Now, taking the reciprocal to find \(d_i\): \[ d_i = -30 \text{ cm} \] ### Step 5: Determine Magnification The magnification (\(m\)) is given by the formula: \[ m = -\frac{d_i}{d_o} \] Substituting the values we have: \[ m = -\frac{-30}{-15} = -2 \] ### Step 6: Determine Image Characteristics - **Image Location**: \(d_i = -30 \text{ cm}\) (the negative sign indicates that the image is on the same side as the object, which is typical for virtual images). - **Magnification**: \(m = -2\) (the negative sign indicates that the image is inverted, and the magnitude indicates that the image is twice the size of the object). - **Real or Virtual**: Since \(d_i\) is negative, the image is virtual. - **Upright or Inverted**: The negative magnification indicates that the image is inverted. ### Final Summary: - **Image Location**: -30 cm - **Magnification**: -2 - **Real or Virtual**: Virtual - **Upright or Inverted**: Inverted