Answer
width: $120~ft$
length: $126~ft$
Work Step by Step
Let $x$ be the width. Then the length is $x+6$. Thus, by the Pythagorean Theorem:
$x^2+(x+6)^2=174^2\\x^2+x^2+12x+36=30276\\2x^2+12x=30240\\x^2+6x-15120=0\\(x-120)(x+126)=0$
Hence, $x=120$ or $x=-126$. However, the dimensions must be positive, so we throw out the negative solution. Thus, the width is $x=120$ and the length is $126$.