一个圆柱的半径为 $(\sqrt{2}-1)\text{ cm}$，体积为 $(12+3\sqrt{2})\pi\text{ cm}^3$。 在不使用计算器的情况下，求其高度 $h\text{ cm}$ 的精确值，表示为 $a+b\sqrt{2}$ 的形式，其中 $a$ 和 $b$ 为整数。

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

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