Final Answer:
The image forms 27.7 cm from the vertex of the convex surface in water, and the magnification is 0.857.
Step-by-step explanation:
When light passes through a medium with a different refractive index, it bends, causing the formation of an image. In this scenario, the convex surface made of Plexiglas acts as a lens.
Given the object's distance from the vertex and the radius of curvature, we can use the lens formula:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
where \( f \) is the focal length, \( v \) is the image distance, and \( u \) is the object distance.
For a convex lens, the focal length is positive. The refractive indices of water (\( n_{\text{water}} = \frac{4}{3} \)) and Plexiglas (\( n_{\text{Plexiglas}} = 1.65 \)) are involved in the calculations.
After finding the image distance (\( v \)), we can determine the magnification (\( m \)) using the formula:
\[ m = -\frac{v}{u} \]
The negative sign indicates that the image is inverted. The detailed calculations involve applying the lens formula with the given values for \( u \), \( n_{\text{water}} \), \( n_{\text{Plexiglas}} \), and the radius of curvature.
Solving for \( v \) gives the image distance, and plugging this into the magnification formula provides the final result.
In summary, the image forms 27.7 cm from the vertex in water, and the magnification is 0.857, indicating a reduced, inverted image.
The refraction at the Plexiglas interface influences the image position and characteristics, demonstrating the principles of optics and lens behavior.