You stand on a straight desert road at night and observe a vehicle approaching. This vehicle is equipped with two small headlights that are 0.659 m apart. At what distance, in kilometers, are you marginally able to discern that there are two headlights rather than a single light source? Take the wavelength of the light to be 545 nm and your pupil diameter to be 4.69 mm.

For a circular aperture, the first minima (n=1) as an angular separation from the peak of the central maxima given by

Sinθ = 1.22λ / d

Where,

d is the aperture or pupil diameter

d = 4.69 mm = 4.69 × 10^-3m

λ is the wavelength

λ = 545 nm = 545 × 10^-9 m

Then,

Sinθ = 1.22λ / d

Sinθ = 1.22 × 545 × 10^-9 / 4.69 × 10^-3

Sinθ = 1.418 × 10^-4 rad

Then, the head light sources have the same angular separation θ from the eye as the image have inside the eye.

For the headlight

Sinθ ≈ light separation / distantce for the eye

Light separation is give as x = 0.659 m

And let the distance of the eye be D

Then,

Sinθ = x / D

Make D subject of formula

D = x / Sinθ

D = 0.695 / 1.418 × 10^-4

D = 4902.316m

To km, 1km = 1000m

D ≈ 4.9 km

volume: 22

density: 7.04

answer: at 24 february, 2014 the population gets twice of its initial population.

explanation:

since we have given that

the population for alpha city, t years after january 1, 2004 is given as

first we find out the initial population, i.e. t=0,

so, our quadratic equation becomes,

according to question, we have said that if the population has twice its initial population, then it becomes

so, time taken to reach this above mentioned population is given by

now, we know that time can't be negative so, we take t=9.149 years.

hence, at 24 february, 2014 the population gets twice of its initial population.