Given perimeter:

$P = b + 2(h - r) + \pi r$

$P = b + 2h - 2r + \pi r$

Where:

$b = 2r$
$r = \frac{1}{2}b$

Thus,

$P = b + 2h - b + \frac{1}{2}\pi b$

$P = 2h + \frac{1}{2}\pi b$

$\dfrac{dP}{db} = 2 \dfrac{dh}{db} + \frac{1}{2}\pi = 0$

$\dfrac{dh}{db} = -\frac{1}{4}\pi$

Light is most if area is maximum:

$A = \frac{1}{2}\pi r^2 + b(h - r)$

$A = \frac{1}{2}\pi (\frac{1}{2}b)^2 + b(h - \frac{1}{2}b)$

$A = \frac{1}{8}\pi b^2 + bh - \frac{1}{2}b^2$

$A = \frac{1}{8}(\pi - 4)b^2 + bh$

$\dfrac{dA}{db} = \frac{2}{8}(\pi - 4)b + b \dfrac{dh}{db} + h = 0$

$\frac{1}{4}\pi b - b - \frac{1}{4}\pi b + h = 0$

$h = b$

∴ breadth = height *answer*