Final answer:
The maximum distance at which the human eye can resolve two headlights of a car is approximately 43.67 meters. The distance between two just-resolvable points held at an arm's length from your eye is approximately 24.1 cm.
Step-by-step explanation:
The maximum distance at which the human eye can resolve two headlights of a car, assuming a pupil diameter of 0.40 cm, is found using the concept of diffraction. The formula for the angle between two just-resolvable points of light is given by:
θ = 1.22 * (λ / d)
where θ is the angle, λ is the average wavelength of light, and d is the diameter of the pupil.
Using the given values, θ = 1.22 * (550 nm / 0.40 cm) = 1.678 degrees.
This angle allows us to calculate the greatest possible distance a car can be from you and still have the headlights be resolved as two separate points. Since the two headlights are 1.3 m apart, we can use trigonometry to find the maximum distance:
d = (1.3 m) / tan(θ) = (1.3 m) / tan(1.678 degrees) = 43.67 m.
Therefore, the greatest possible distance a car can be from you while still being able to resolve its two headlights is approximately 43.67 meters.
The distance between two just-resolvable points held at an arm's length (0.8 m) from your eye can also be calculated using the same angle:
d = (0.8 m) * tan(θ) = (0.8 m) * tan(1.678 degrees) = 0.0241 m = 24.1 cm.
So, the distance between two just-resolvable points held at an arm's length from your eye is approximately 24.1 cm.