Answer
$V(w)= \frac{1}{2}w^{3}, w\gt 0$
Work Step by Step
We are told that the box has a height half the width ($h=0.5w$). We also know that the base is square ($l=w$). Thus the volume is:
Volume=$l*w*h=w*w*0.5w=0.5w^3=\frac{1}{2}w^3$
We restrict $w>0$ to avoid negative sides.
