Answer
$12.17m$
Work Step by Step
$r_{x}=\left( a_{x}+b_x\right) \widehat {i}$
$r_{y}=\left( a_{y}+b_y\right) \widehat {j}$
Lets calculate $a_{x}\widehat {i}$ ,$b_{x}\widehat {i}$. ,$a_{y}\widehat {j}$ and $b_{y}\widehat {j}$
$a_x=a\times \cos \theta _{1}=10.0m\times \cos 30= 5 \sqrt 3 m;$
$b_x=10.0m\times \cos \left( \theta _{2}+\theta _{1}\right) =10.0m\times \cos \left( 105+30\right) =10.0m\times \cos 135=-5\sqrt {2}m$
$\Rightarrow r _{x}=a_{x}+bx=5\left( \sqrt {3}-\sqrt {2}\right) \approx 1.59m$
$a_y=a\times \sin \theta _{1}=10.0m\times \sin 30= 5 m;$
$b_y=10.0m\times \sin \left( \theta _{2}+\theta _{1}\right) =10.0m\times \sin \left( 105+30\right) =10.0m\times \sin 135=5\sqrt {2}m$
$\Rightarrow r _{y}=a_{y}+b_y=5\left( 1+\sqrt {2}\right) \approx 12.07m$
$r=\sqrt {r^{2}_{x}+r^{2}_{y}}\approx 12.17m$
