Answer:
5. 10 cm
6. 16 cm
Step-by-step explanation:
Thin lens problems boil down to two formulas, one for image position, and one for image size (and inversion).
If "o" and "i" are the object and image positions measured in the same direction from the lens, and "f" is the focal length, then ...
1/o - 1/i = 1/f . . . . . . i < 0 for images on the opposite side of the lens
i/o = magnification . . . . . "m" < 0 for inverted image
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5. For distances in cm, you have o = 5.5, i = 12, and you want to find f.
1/5.5 - 1/12 = 1/f
(12 -5.5)/(5.5·12) = 1/f . . . . add the fractions
f = 66/6.5 ≈ 10 . . . . . . . . . invert the sum
The focal length is about 10 cm.
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6. i/o = -3 . . . . . . the image is 3 times as tall and inverted
So, i = -3o
And f = 12 centimeters. For distance o in centimeters, we have ...
1/o - 1/(-3o) = 1/12
(-3o -o)/(-3o^2) = 1/12 . . . . . add the fractions
4/(3o) = 1/12 . . . . . . . . . . . . . simplify
48/3 = o . . . . . . . . . . . . . . . . multiply by 12o
16 = o . . . . . simplify
The object is 16 cm from the lens.