We get that:
[tex]\begin{gathered} A=l\cdot w \\ 2l+2w=172\rightarrow l+w=86\rightarrow l=86-w \end{gathered}[/tex]so we get that the area is
[tex]A=(86-w)w=86w-w^2[/tex]so we get that the width of the rectangle with largest area is
[tex]w=-\frac{86}{-2}=43[/tex]and the length is
[tex]l=86-43=43[/tex]so the dimensions are 43 of length and 43 of width
