Assume the distance between the object and screen is x, then object distance (between object and lens) plus image distance (between screen and lens) equals x:
x=i1+o1=i2+o2 (0)
for both cases.
Also, we know that magnification is the image distance divided by object distanceб or image height divided by object height:
m1=o1i1, m1=hohi1=ho4→o1i1=h4. m2=o2i2, m2=hohi2=ho16→o2i2=h16.
Divide one by another:
i2o1i1o2=41⇄o1i1=4o2i2. (1)
Multiply one by another:
o1o2i1i2=h264. (2)
Moreover, according to thin lens equation, we have
f1=i11+o11=i21+o21, or i1o1i1+o1=i2o2i2+o2⇄o2i1=o1i2. (3) Plug equation (1) in (2):
4o22i22=h264. (4) Plug equation (3) in (2):
o12i22=h264. (5) Since right parts of (4) and (5) are equal, the left pars are equal as well:
4o22i22=o12i22→o1=2o2.Now plug this o-equation in (3):
o2i1=2o2i2→i1=2i2. (6)
Now look at the equation (0) in the very beginning and plug (6) into it:
2i2+2o2=i2+o2→i2=2o2. (7)
Finally, it's time to plug (7) in (4):
4o22(2o2)2=h264, 1=h264→h=8 cm.
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