Answer:
Step-by-step explanation:
First, we need to calculate the angle of separation between the two numbers on the plate as seen from the satellite:
tan(theta) = opposite/adjacent
tan(theta) = (5.5 cm)/160,000 m
theta = tan^-1(5.5/160,000) = 0.002 degrees
Next, we can use the small angle approximation to relate this angle to the diameter of the camera's aperture:
sin(theta) ≈ theta ≈ (D/2)/f
where D is the diameter of the aperture and f is the focal length.
Rearranging, we get:
D ≈ 2*theta*f
We can assume that the focal length of the camera is much larger than the distance to the satellite, so we can use the thin lens equation to relate f to the distance to the satellite:
1/f = 1/d_o + 1/d_i
where d_o is the distance from the lens to the object (i.e. the distance from the satellite to the plate) and d_i is the distance from the lens to the image (i.e. the distance from the lens to the camera sensor).
Since the satellite is so far away, we can assume that d_o is equal to the distance to the satellite (160 km). We want to focus on an object that is essentially at infinity, so we can set d_i equal to the focal length (i.e. a distant object will be in focus at the focal plane of the lens).
1/f = 1/160,000,000 + 1/f
1/f - 1/f = 1/160,000,000
f = 160,000,000 meters
Now we can substitute f and theta into our equation for D:
D ≈ 2*theta*f
D ≈ 2*(0.002)*(160,000,000)
D ≈ 640 meters
Therefore, the diameter of the camera's aperture must be at least 640 meters in order to resolve the numbers on the plate from an altitude of 160 km.