Diffraction Grating problem (waves and optics)

[pondzo]

Homework Statement

Visible light passes through a diffraction grating that has 900 slits per centimeter, and the interference pattern is observed on a screen that is 2.58m from the grating. In the first-order spectrum, maxima for two different wavelengths are separated on the screen by 3.16 mm . What is the difference between these wavelengths?

Homework Equations

d*sinΘ=mλ

The Attempt at a Solution

d = 1/90000 = 1.111*10^-5

d*sin(Θ₁)=λ₁ and d*sin(Θ₂)=λ₂

using subtraction d*ΔsinΘ=Δλ (and using small angle approx d*ΔΘ=Δλ)

But I am not sure how to find ΔΘ (or how even to visualize ΔΘ)

thanks in advance.

 

[BvU]

This help ?

On the left is the grating, on the right is the screen.
(grating details not shown on this scale)

 

[pondzo]

Thank you for making it clearer! i was originally thinking this but i didn't think i could calculate it since i didn't have all of the distance of the opposite side (the screen).

Is this the correct way to go about it?

Sin(ΔΘ) = sin(Θ2) - sin(Θ1)
= x/2.58 + (3.16*10^-3-x)/2.58 (where x is the total distance on the screen)
= (3.16*10^-3)/2.58
ΔΘ=0.001225

and λ=1.225*10^-3*1.11*10^-5
=1.36*10^-8