Answer
The weight of the box is 226.2 lb
Work Step by Step
Let $a = 150~lb$
Let $b = 114~lb$
Let angle $\theta$ be the angle between these two vectors. Then $\theta = 180-62.4^{\circ}-54.9^{\circ} = 62.7^{\circ}$
We can use the parallelogram rule to find $c$, which is equal in magnitude to the weight of the box:
$c = \sqrt{a^2+b^2+2ab~cos~\theta}$
$c = \sqrt{(150~lb)^2+(114~lb)^2+(2)(150~lb)(114~lb)~cos~62.7^{\circ}}$
$c = \sqrt{51181.8~lb^2}$
$c = 226.2~lb$
The weight of the box is 226.2 lb