2011SP-I-11

For $s=2$ and $C=356$, we have

$\begin{array}{rcl}
356 & = & 2k_1+4k_2 \\
178 & = & k_1+2k_2~\ldots \unicode{x2460}
\end{array}$

For $s=5$ and $C=1~250$, we have

$\begin{array}{rcl}
1250 & = & 5k_1+25k_2 \\
250 & = & k_1+5k_2~\ldots \unicode{x2461}
\end{array}$

$\unicode{x2461} – \unicode{x2460}$, we have

$\begin{array}{rcl}
72 & = & 3k_2 \\
k_2 & = & 24~\ldots \unicode{x2462}
\end{array}$

Sub. $\unicode{x2462}$ into $\unicode{x2460}$, we have

$\begin{array}{rcl}
178 & = & k_1+2(24) \\
k_1 & = & 130
\end{array}$

$\therefore C=130s+24s^2$.

For $s=6$,

$\begin{array}{rcl}
C & = & 130(6)+24(6)^2 \\
& = & 1~644
\end{array}$

$\begin{array}{rcl}
130s+24s^2 & = & 539 \\
24s^2+130s-539 & = & 0 \\
(4s-11)(6s+49) & = & 0 \\
\end{array}$

$\therefore s=\dfrac{11}{4}$ or $s=-\dfrac{49}{6}$.

Since perimeter must be positive, then the perimeter is $\dfrac{11}{4}$.