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Perimeter $= d$

$4x + 2y + \pi x = d$

Area, $A = xy + \frac{1}{2} \pi x^2$

$A = x(\frac{d-4x-\pi x}{2}) + \frac{1}{2} \pi x^2$

$A = \frac{d}{2}x - 2x^2$

$\frac{dA}{dx} = \frac{d}{2} - 4x$

Let $\frac{dA}{dx} = 0 \Rightarrow, x =\frac{d}{8}$

Since $\frac{d^2A}{dx^2} = -4 \textless 0$, A is maximum when $x = \frac{d}{8}$

$A = 0.03125d^2$

$\therefore, k = 0.03125 = \frac{1}{32}$