the lens of a slide projector focuses an image of height 1.5 m on a screen placed 9 m from the projector. If the height of the picture on the slide is 6.5vm determine the distance between the slide (picture) and the lens
To solve this problem, we can use the thin lens formula:
1/f = 1/do + 1/di
Where:
f = focal length of the lens
do = object distance (distance between the lens and the slide)
di = image distance (distance between the lens and the screen)
Given:
Height of the image on the screen, hi = 1.5 m
Distance between the projector and the screen, di = 9 m
Height of the picture on the slide, ho = 6.5 cm = 0.065 m
We can use the properties of similar triangles to relate the heights of the object, image, and their distances from the lens:
ho/do = hi/di
Plugging in the given values, we get:
0.065 m/do = 1.5 m/9 m
Solving for do:
do = (0.065 m * 9 m) / 1.5 m
do = 0.39 m
Now, we can use the thin lens formula to find the focal length of the lens:
1/f = 1/do + 1/di
1/f = 1/0.39 + 1/9
1/f = 2.5641 + 0.1111
1/f = 2.6752
f = 1 / 2.6752
f ≈ 0.374 m
Therefore, the distance between the slide (picture) and the lens is approximately 0.39 meters.