What are some hints for solving Geometrical Optics problems?

[Perses]

I'm having some trouble with a couple of Optics questions, any hints would be much appreciated!

THe first quetion states that a concave mirror forms an image on a screen twice as large as the object. The object and the screen are then moved such that the new image produced is 3x's the size of the object. THe screen was moved 75cm, how far was the object moved and what is the focal length of the mirror?

I'm afraid I'm not sure what to do; I thought i could somehow use the magnification formula (m=-s'/s) and compare the two different scenarios but i couldn't seem to isolate for only the object distance travelled.

The second question states that a glass hemisphere is silvered over it's curved surface. A small air bubble in the glass is located on the central axis through the hemisphere 5cm away from the plane surface. The radius of curvature of the spherical surface is 7.5cm, and the glass has an index of 1.5. Looking along the axis into the plane surface, one sees two images of the bubble. How do they arise and where do they appear?

I'm fairly certain as to how they appear. The first is formed directly from a ray that is refracted by the plane surface. The next is formed when a ray is reflected off of the inner front surface of the plane mirror and then reflected off of the rear surface of the mirror and finally refracted out into the air.
However, I'm not quite sure how to find there heights. Any help on where to begin would be much appreciated.

Thanks for your time guys.

 

[lightgrav]

the image must be IN FOCUS,
as well as being the right magnification
(that's why they had to move object AND screen)

The second bubble image will be formed by the light
that goes first to the curved mirror surface, being
reflected to pass through the glass
and refract at the plane surface. [ignore reflection at plane]

the first bubble image will appear closer to the flat (by 1/n);
the second WOULD've formed at 1/d_im = 1/f - 1/d_ob
except that it runs out of glass halfway to that 15cm.
So Snell refracts the actual focus (small angle approx)
to only 2/3 of the 7.5 cm distance in air.

 

[Perses]

Thnx a lot
I'm having a bit of trouble with another problem

To determine the refractive index of a transparent plate of glass, a microscope is first focused on a tiny scratch in the upper surface, and the barrel position is recoreded. Upon further lowering the microscope barrel by 1.87mm, a focused image of the scratch is seen again. The plate thickness is 1.50mm. What is the refractive index of the glass?

I believe the light rays reflect off of the back surface of the plane and are then reflected back and refracted, this is how the 2nd image is produced. However, i cannot seem to figure out the refractive index. Can anyone lend me a hand?