Single Slit Diffraction and Monochromatic light

[jones268]

Monochromatic light with a wavelength of 419 nm passes through a single slit and falls on a screen 88 cm away. If the distance of the first-order dark band is 0.29 cm from the center of the pattern, what is the width of the slit?

My knowns are as follows:
L= 88 cm (distance from slit to screen)
λ= 419 nm
WCM= 0.58 cm (width of the central max, I assumed it was twice the distance from the center of the patter to the first order dark band)
w=? unknown width of the slit

I thought I should use the following equation:

WCM=[(2)(λ)(L)]/[(square root of : (w^2 - λ^2))]

I plugged the numbers into the equation and solved for w, but came up with the wrong answer, I'm not quite sure what I'm doing wrong...

 

[dacruick]

you should be able to solve for an angle using the distance of the first order band. After that, there will be a formula that relates all of your other components to theta and slit width.

 

[Stonebridge]

Shouldn't you be using the formula for the direction to the first minimum
sin θ = nλ/a where a is the slit width?

 

[jones268]

Thanks stonebridge for the help, I never thought of using that equation for L. But I'm still coming up short on the right answer. Instead of 0.0127145 cm as my answer, I'm coming up with 0.0165528450 cm. :/ This is so frustrating.

 

[dacruick]

use the formula that stonebridge just gave you. I got the right answer. make sure that you calculate your theta using arctan(.29/88)