Answer
The shortest length of cable needed is 3835ft long
Work Step by Step
To find the length of the cable, one must first find $x$ (see picture). $x$ can be found using the sine function:
sin$(74º)=\frac{3400}{x}$
$x=\frac{3400}{sin(74º)}\approx3537$ft
Now, one has all the data to find the length of the cable (variable $l$ will be used) using the law of cosines:
$l^2=800^2+3537^2-2(800)(3537)\cdot $cos$(106º)$
$l^2=640,000+12,510,369-5,659,200\cdot (-0.28)$
$l^2=13,150,369+1,559,887$
$l=\sqrt{14,710,256}$
$l \approx 3835$ft
