You can use the Law of Sines (sina/A=sinb/B=sinc/C)
We are told that AB=4in so
AD/sin45=AB/sin30
AD/sin45=4/sin30
AD=4sin45/sin30
Since sin45=1/√2 and sin30=1/2
AD=4(1/√2)/(1/2)
AD=8/√2, when rationalized
AD=8√2/2
AD=4√2 in
AD≈5.66 in (to nearest hundredth of an inch)