图中显示了一个高度为 x cm 的长方体。 长方体的宽比它的高多 6 cm。
长方体的长是它宽的两倍。 长方体的对角线 AG 的长度为 24 cm。

1. Express all three dimensions of the cuboid in terms of $x$.

Height $=x$

Width $=x+6$

Length $=2(x+6)=2x+12$

2. Form the equation using the space diagonal of the cuboid.

By Pythagoras' theorem in 3 dimensions,
$AG^2=(\text{length})^2+(\text{width})^2+(\text{height})^2.$
Since $AG=24$,
$24^2=(2x+12)^2+(x+6)^2+x^2.$

3. Expand the squares and simplify the equation.

$576=(2x+12)^2+(x+6)^2+x^2$
$576=(4x^2+48x+144)+(x^2+12x+36)+x^2$
$576=6x^2+60x+180$

4. Rearrange to show the required quadratic equation.

$6x^2+60x+180-576=0$
$6x^2+60x-396=0$
Divide by $6$:
$x^2+10x-66=0$

1. Identify the coefficients and substitute them into the quadratic formula.

For $x^2+10x-66=0$, $a=1$, $b=10$, $c=-66$.

Using
$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$
$x=\frac{-10\pm\sqrt{10^2-4(1)(-66)}}{2(1)}.$

2. Simplify the discriminant and the exact expression for the two roots.

$x=\frac{-10\pm\sqrt{100+264}}{2}$
$x=\frac{-10\pm\sqrt{364}}{2}.$

3. Calculate both numerical solutions.

$x=\frac{-10+\sqrt{364}}{2}=4.355\ldots$
$x=\frac{-10-\sqrt{364}}{2}=-14.355\ldots$

4. Round both solutions to two decimal places.

Correct to two decimal places,
$x=4.36\quad\text{or}\quad x=-14.36.$

1. Use the physically valid value of $x$ and find the side opposite angle $GAF$.

From part (b), take the positive value
$x=4.36$
since the height of a cuboid cannot be negative.

The width is
$x+6=4.36+6=10.36\text{ cm}.$
In triangle $AFG$, $FG$ is the width of the cuboid, so
$FG=10.36\text{ cm}.$

2. Apply a trigonometric ratio in right-angled triangle $AFG$ to find angle $GAF$.

In right-angled triangle $AFG$, the hypotenuse is
$AG=24\text{ cm}.$

For angle $GAF$,
$\sin \angle GAF=\frac{FG}{AG}$
$\sin \angle GAF=\frac{10.36}{24}.$

Hence
$\angle GAF=\sin^{-1}\left(\frac{10.36}{24}\right)=25.6^\circ\text{ correct to 1 decimal place}.$