A cylinder has a radius of $(\sqrt{2}-1)\text{ cm}$ and a volume of $(12+3\sqrt{2})\pi\text{ cm}^3$. Find, without using a calculator, the exact value of its height, $h\text{ cm}$, in the form $a+b\sqrt{2}$, where $a$ and $b$ are integers.

1. Set up the equation for the volume of the cylinder

Volume of cylinder, $V = \pi r^2 h$
$(12+3\sqrt{2})\pi = \pi (\sqrt{2}-1)^2 h$

2. Expand the squared term and make $h$ the subject

$12+3\sqrt{2} = (2 - 2\sqrt{2} + 1) h$
$12+3\sqrt{2} = (3 - 2\sqrt{2}) h$
$h = \frac{12+3\sqrt{2}}{3-2\sqrt{2}}$

3. Rationalise the denominator to find $h$ in the required form

$h = \frac{12+3\sqrt{2}}{3-2\sqrt{2}} \times \frac{3+2\sqrt{2}}{3+2\sqrt{2}}$
$h = \frac{36 + 24\sqrt{2} + 9\sqrt{2} + 6(2)}{3^2 - (2\sqrt{2})^2}$
$h = \frac{36 + 33\sqrt{2} + 12}{9 - 8}$
$h = 48 + 33\sqrt{2}$