A 17.1 cm focal-length lens is used to project 35 mm slides onto a screen 4.77 m from the projector lens. What must be the width of the screen ( m) if the image of the slide is to just fill the screen? The width of the image on the slide is 34.3 mm.
Get the distance from lens to slide , do, by using the equation
1/di + 1/do = 1/f
where f = 17.1 cm and image distance di = 477 cm
I get 1/do = 0.05638
do = 17.7 cm
The magnification is di/do = 477/17.7 = 27.0
27 times the slide picture width is 924 mm or 0.924 meter
That is how wide the screen must be.
To find the width of the screen, we need to use the formula for the magnification of a lens:
Magnification (M) = - image distance / object distance
In this case, the object distance is the distance from the lens to the slide (which is the focal length of the lens) and the image distance is the distance from the lens to the screen.
Given:
Focal length of the lens (f) = 17.1 cm
Distance from the lens to the screen (d) = 4.77 m
Width of the image on the slide (W) = 34.3 mm
First, we need to convert the given values to the same units. Let's convert all distances to meters and the width of the image to meters:
17.1 cm = 0.171 m
4.77 m = 4.77 m
34. 3 mm = 0.0343 m
Now, we can substitute the values into the magnification formula:
M = - d / f
Rearranging the formula to solve for d:
d = -M * f
Substituting the given values:
d = -(-0.0343 / 0.171) * 0.171
= 0.0343 m
The image distance from the lens to the screen is 0.0343 m.
To find the width (W) of the screen, we can use the similar triangles principle:
W / d = W' / f
Where W is the width of the screen and W' is the width of the image on the screen.
Substituting the given values:
W / 0.0343 = 0.0343 / 0.171
Solving for W:
W = 0.0343 * 0.0343 / 0.171
= 0.00694 m
Therefore, the width of the screen must be approximately 0.00694 meters (or 6.94 mm) to ensure that the image of the slide fills the screen.