如图 1 所示，一个实体模型是通过将一个三棱柱连接到一个长方体上而制成的。该模型的体积为 $28224\text{ cm}^3$，表面积为 $S\text{ cm}^2$。

1. Find the volume of the rectangular cuboid in terms of $x$ and $h$.

The cuboid has width $12x$, height $x$, and depth $h$.
$\text{Volume of cuboid} = 12x \times x \times h = 12x^2h$

2. Find the volume of the triangular prism in terms of $x$ and $h$.

The base of the triangular face $GE$ is $12x - 2x - 2x = 8x$.
The triangle is isosceles with slant edges $5x$. Using Pythagoras' theorem, the perpendicular height of the triangle is:
$\text{Height} = \sqrt{(5x)^2 - \left(\frac{8x}{2}\right)^2} = \sqrt{25x^2 - 16x^2} = \sqrt{9x^2} = 3x$
$\text{Area of triangular face} = \frac{1}{2} \times 8x \times 3x = 12x^2$
$\text{Volume of triangular prism} = 12x^2 \times h = 12x^2h$

3. Formulate an equation using the total volume and solve for $h$.

$\text{Total volume} = 12x^2h + 12x^2h = 24x^2h$
Given that the total volume is $28224\text{ cm}^3$:
$24x^2h = 28224$
$h = \frac{28224}{24x^2}$
$h = \frac{1176}{x^2}$

1. Determine the expression for $h$ from the given volume in part (i).

2. Formulate the total surface area $S$ in terms of $x$.

Summing the outer surfaces:

Substitute $h = \frac{1176}{x^2}$:

3. Differentiate $S$ with respect to $x$ and equate to zero to find the stationary point.

For a minimum, set $\frac{\mathrm{d}S}{\mathrm{d}x} = 0$:

4. Solve for $x$.

5. Use the second derivative test to justify that $S$ is a minimum.

When $x = 7$:

Since $\frac{\mathrm{d}^2S}{\mathrm{d}x^2} > 0$, the surface area $S$ is a minimum when $x = 7$ (Shown).