Oblique Cylinder 2

$V = \pi r^2 H = b^2 \implies H = \dfrac{b^2}{\pi r^2}$ and $A = 2\pi rs = b \implies s = \dfrac{b}{2\pi r}$

$\implies Y(r) = y^2(r) = s^2(r) - H^2(r) = \dfrac{b^2}{\pi^2}(\dfrac{1}{4 r^2 } - \dfrac{b^2}{r^4}) \implies \dfrac{dY}{dr} = \dfrac{b^2}{\pi^2}(\dfrac{8 b^2 - r^2}{2r^5})$ $\:\ r > 0 \implies \boxed{r = 2\sqrt{2} b} \implies$ $s = \dfrac{1}{4\sqrt{2}\pi}$ and $H = \dfrac{1}{8\pi}$ and $y = \sqrt{s^2 - H^2} = \dfrac{1}{8\pi} = H$ .

For $0 < b < \dfrac{3}{5} \implies \dfrac{d^2Y}{dr^2}|_{r = 2\sqrt{2} b} = \dfrac{1}{128\pi^2}(3 - 5b) > 0 \implies$ min at $r = 2\sqrt{2} b$ .

Let $m\angle{PTR} = \lambda \implies \cos(\lambda) = \dfrac{y(r)}{s(r)} = \dfrac{1}{\sqrt{2}}$ and $\theta = 180 - \lambda \implies \cos(\theta) = \cos(180 - \theta) = -\cos(\lambda) = -\dfrac{1}{\sqrt{2}} \implies \boxed{\theta = 135^\circ}$

$\implies d^2 = s^2 + 4r^2 + 4rs \cos(\lambda) = \dfrac{1}{32\pi^2} + 32 b^2 + \dfrac{\sqrt{2} b}{\pi} =$ $\dfrac{32^2\pi^2 b^2 + 32\sqrt{2}\pi b + 1}{32\pi^2} \implies \boxed{d = \dfrac{\sqrt{32^2\pi^2 b^2 + 32\sqrt{2}\pi b + 1}}{4\sqrt{2}\pi}}$

$d^2 = \dfrac{32^2\pi^2 b^2 + 32\sqrt{2}\pi b + 1}{32\pi^2} = \dfrac{b}{2\pi} (6\pi b + 3\sqrt{2}) \implies$ $32^2\pi^2 b^2 + 32\sqrt{2}\pi b + 1 = 32\pi^2 b(\dfrac{64\pi b + 3\sqrt{2}}{2\pi}) =$ $32\pi^2(32 b^2 + \dfrac{3}{2\pi}\sqrt{2} b) = 32^2\pi^2b^2 + 48\sqrt{2}\pi b \implies$ $16\sqrt{2}\pi b = 1 \implies b = \dfrac{1}{16\sqrt{2}\pi} \approx \boxed{0.014067}.$