The diagram shows a cuboid with height x cm. The width of the cuboid is 6 cm more than its height.
The length of the cuboid is twice its width. The length of the diagonal, AG, of the cuboid is 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
$\angle GAF=\sin^{-1}\left(\frac{10.36}{24}\right)=25.6^\circ\text{ correct to 1 decimal place}.$