An optics manufacturer is designing a portable projector that uses a single thin converging lens to project enlarged images of 2.0cm-tall slides onto a white screen. The distance between the slide and the screen must be fixed at 1.20m due to the projector's compact housing.
(a) Derive an algebraic expression for the image distance si in terms of the object distance so and the focal length f.
(b) The manufacturer requires a screen image height of at least 0.30m. In terms of so and si, write an inequality that represents this design constraint.
(c) Using your expression from part (a) and the magnification relationship, determine the range of object distances so that meet the image size requirement while still producing a real image on the screen. Show all steps clearly and include units.
(d) From your result in part (c), choose a convenient object distance within the allowable range and calculate the corresponding focal length the lens must have. Finally, sketch a qualitative ray diagram that confirms your numerical findings and comment on any practical considerations the manufacturer should keep in mind (e.g., lens diameter, slide placement tolerance).