一个长方体的底面是边长为 $(3+\sqrt{2})\text{ cm}$ 的正方形，高为 $h\text{ cm}$。该长方体的体积为 $(19-3\sqrt{2})\text{ cm}^3$。在不使用计算器的情况下，求出 $h$ 的表达式，并化为 $(a+b\sqrt{2})$ 的形式，其中 $a$ 和 $b$ 为整数。

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

$\text{Volume} = \text{Base Area} \times \text{Height}$
$(3+\sqrt{2})^2 \times h = 19-3\sqrt{2}$

2. Expand the base area

$(3+\sqrt{2})^2 = 3^2 + 2(3)(\sqrt{2}) + (\sqrt{2})^2$
$= 9 + 6\sqrt{2} + 2$
$= 11 + 6\sqrt{2}$

3. Express $h$ as a fraction

$h = \frac{19-3\sqrt{2}}{11+6\sqrt{2}}$

4. Rationalise the denominator to find $h$

$h = \frac{19-3\sqrt{2}}{11+6\sqrt{2}} \times \frac{11-6\sqrt{2}}{11-6\sqrt{2}}$
$= \frac{19(11) - 19(6\sqrt{2}) - 3\sqrt{2}(11) + 3\sqrt{2}(6\sqrt{2})}{11^2 - (6\sqrt{2})^2}$
$= \frac{209 - 114\sqrt{2} - 33\sqrt{2} + 18(2)}{121 - 36(2)}$
$= \frac{209 - 147\sqrt{2} + 36}{121 - 72}$
$= \frac{245 - 147\sqrt{2}}{49}$
$= 5 - 3\sqrt{2}$