Coherent light of frequency 6.34 x 10^14 Hz passes through two thin slits and falls on a screen 82.0 cm away. You observe that the third bright fringe occurs at ± 3.09 cm on either side of the central bright fringe.
(a) How far apart are the two slits?
(b) At what distance from the central bright fringe will the third dark fringe occur?

a) d = 3.76 10⁻⁵ m , b)  y = 3.61 10⁻² m

Explanation:

The interference phenomenon is described by the expression

         d sin θ = m λ

Where d is the separation of the slits,  λ the wavelength and m an integer indicating the interference number

Let's apply to this case, as we observe the third order m = 3

a) Let's use trigonometry to find the angles

           tan θ = y / x

The phenomena of interference and diffraction angles are very small, so

          tan  θ = sin θ / cos θ = sin θ

          sin θ = y / x

We replace

         d y / x = m  λ

         d = m  λ x / y

The frequency of the light is

         c =  λ f

          λ= c / f

Let's calculate

          λ = 3 10⁸ / 6.34 10¹⁴

           λ = 4.73 10⁻⁷ m

          d = 3  4.73 10⁻⁷  0.820 / 0.0309

          d = 376.56 10⁻⁷ m

          d = 3.76 10⁻⁵ m

b) for destructive interference we must add ½ Lam

            d sin θ = (m + ½) λ

            d y / x = (m + ½) λ

            y = (m + ½) λ x / d

Let's calculate

            y = (3+ ½)  4.73 10⁻⁷  0.820 / 3.76 10⁻⁵

            y = 3.61 10⁻² m

the first, second, and fifth options

im not to sure what it is