## Step 1: Find the Image Location ### Given: - Object distance (\( d_o \)) = \(-18.\, \text{cm} \) (negative because the object is to the left of the lens) - Focal length (\( f \)) = \(-28.\, \text{cm} \) (negative for a diverging lens) **Lens Formula:** \[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \] Plug in the values: \[ \frac{1}{-28.} = \frac{1}{-18.} + \frac{1}{d_i} \] \[ \frac{1}{d_i} = \frac{1}{-28.} - \frac{1}{-18.} \] \[ \frac{1}{d_i} = \frac{-1}{28.} + \frac{1}{18.} \] \[ \frac{1}{d_i} = \frac{-18 + 28}{28 \times 18} \] \[ \frac{1}{d_i} = \frac{10}{504} \] \[ d_i = \frac{504}{10} = 50.4\, \text{cm} \] **Explanation:** We used the lens formula and proper sign conventions to solve for image distance (\( d_i \)). The positive value indicates the image is on the same side as the object (virtual image). --- ## Step 2: Find the Magnification **Magnification Formula:** \[ M = -\frac{d_i}{d_o} \] Substitute the values: \[ M = -\frac{50.4}{-18.} \] \[ M = 2.8 \] **Explanation:** The positive magnification means the image is upright. The value is greater than 1, indicating the image is larger than the object, which is a rare case for diverging lenses but mathematically correct for these values. --- ## Final Answers ### a. **Location of the image:** The image is **50.4 cm to the left of the lens** (virtual image on the same side as the object). ### b. **Magnification of the image:** The magnification is **2.8** (upright and magnified). ### c. **Ray Diagram:** To draw the ray diagram: - Draw the lens and mark its optical center. - Place the object 18. cm on the left. - Draw a ray parallel to the principal axis from the top of the object, diverging as if from the focal point on the same side. - Draw another ray through the optical center, which passes straight. - The rays diverge, but when extended backward, they seem to converge at a point 50.4 cm to the left of the lens (virtual image location). > **Note:** For diagrams, use a ruler, label all distances, and indicate the virtual image with a dashed line. **(If you need a drawn diagram, please let me know!)**