Answer
$12.07m$
Work Step by Step
If
$\overrightarrow {r}=\overrightarrow {a}+\overrightarrow {b}$
$\Rightarrow r=(a_x\widehat {i}+a_y\widehat {j}+a_{z}\widehat {k})+(b_x\widehat {i}+b_y\widehat {j}+b_{z}\widehat {k})=(a_x +b_x)\widehat {i} +(a_y+b_y)\widehat {j}+(a_{z}+b_{z})\widehat {k}$
$r_{y}=\left( a_{y}+b_y\right) \widehat {j}$
Lets calculate $a_{y}\widehat {j}$ and $b_{y}\widehat {j}$
$a_y=a\times \sin \theta _{1}=10.0m\times \sin 30=5m;$
$b_x=10.0m\times \sin \left( \theta _{2}+\theta _{1}\right) =10.0m\times \sin \left( 105+30\right) =10.0m\times \sin 135=5\sqrt {2}m$
$\Rightarrow r _{x}=a_{x}+b_{x}=5+5 \sqrt {2} \approx 12.07m$
