Total cost:

$k = mx + n(2y + x)$

$k = mx + 2ny + nx$

$k - (m + n)x = 2ny$

$y = \dfrac{k}{2n} - \dfrac{m + n}{2n}x$

Area:

$A =xy$

$A = x\left( \dfrac{k}{2n} - \dfrac{m + n}{2n}x \right)$

$A = \dfrac{k}{2n}x - \dfrac{m + n}{2n}x^2$

$\dfrac{dA}{dx} = \dfrac{k}{2n} - \dfrac{m + n}{n}x = 0$

$\dfrac{k}{2n} = \dfrac{m + n}{n}x$

$x = \dfrac{k}{2(m + n)}$

$y = \dfrac{k}{2n} - \dfrac{m + n}{2n} \, \left[ \dfrac{k}{2(m + n)} \right]$

$y = \dfrac{k}{4n}$

$A = \dfrac{k}{2(m + n)} \times \dfrac{k}{4n}$

$A = \dfrac{k^2}{8n\,(m + n)} \, \text{ yard}^2$ *answer*