A 35-mm camera equipped with a 40-mm focal length lens is used to photograph a tree 17 m tall. If a 32-mm high image of the tree on the CCD sensor is required, how far should the camera lens be from the tree?
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To solve this problem, we can use the thin lens equation, which relates the object distance (distance from the lens to the object), the image distance (distance from the lens to the image), and the focal length of the lens.
The thin lens equation is given by:
1/f = 1/d_o + 1/d_i
where f is the focal length, d_o is the object distance, and d_i is the image distance.
In this case, the object distance (d_o) is the distance from the camera lens to the tree and the image distance (d_i) is the distance from the camera lens to the sensor.
Given:
f = 40 mm = 0.04 m
Height of the tree = 17 m
Height of the image on the sensor = 32 mm = 0.032 m
Now, to find the object distance, we can use the magnification equation:
m = h_i / h_o
where m is the magnification, h_i is the height of the image, and h_o is the height of the object.
Substituting the given values:
m = 0.032 m / 17 m
Next, we can use the magnification equation in terms of the object and image distances:
m = -d_i / d_o
Since the negative sign is used to indicate that the image is inverted, we can ignore it for this problem.
Substituting the value of magnification:
0.032 m / 17 m = -d_i / d_o
Now, we can rearrange the equation to solve for the object distance (d_o):
d_o = -d_i * 17 m / 0.032 m
Substituting the given value of image distance (d_i):
d_o = -(0.032 m) * 17 m / 0.032 m
Simplifying the equation:
d_o = -0.032 m * 17 m / 0.032 m
= -17 m
Since we want to find the distance from the camera lens to the tree (d_o), which is a positive distance, we can take the absolute value of d_o:
d_o = |d_o| = 17 m
Therefore, the camera lens should be placed 17 meters away from the tree in order to obtain a 32-mm high image of the tree on the CCD sensor.
To solve this problem, we can use the concept of similar triangles.
First, let's set up the ratio of the height of the tree to the height of the image:
Tree height / Image height = Distance from lens to tree / Focal length of lens
Plugging in the given values:
17 m / 32 mm = Distance from lens to tree / 40 mm
Now, let's convert the units to be consistent:
17 m / (32 mm/1000) = Distance from lens to tree / (40 mm/1000)
17 m / 0.032 m = Distance from lens to tree / 0.04 m
Next, let's solve for the unknown distance using cross-multiplication:
0.032 m * Distance from lens to tree = 17 m * 0.04 m
Simplifying the equation:
Distance from lens to tree = (17 m * 0.04 m) / 0.032 m
Distance from lens to tree = 21.25 m
Therefore, the camera lens should be approximately 21.25 meters away from the tree.