Golden ratio 1: Cube with Φ

Φ represents the golden ratio, or $\frac{1+√5}{2}$ . $\frac{1+√5}{2}$ ^3, using the special product property for $(a+b)^3$ , simplifies to $2+\sqrt{5}$ . We will use this information later. It is given that the volume is $54+\sqrt{3645}$ . $\sqrt{3645} = \sqrt{(729*5)}$ , which is $27\sqrt{5}$ . Now the expression simplifies to $54+27\sqrt{5}$ , and we can factor out $27$ to get $27(2+\sqrt{5})$ . We already found out that $Φ^3 = 2+\sqrt{5}$ . As a result, the side length of the cube is the cube root of $27(2+\sqrt{5})$ , or $3Φ$ . Since the space diagonal of a cube is the side length times $\sqrt{3}$ , $(3\sqrt{3})Φ$ is the length of the space diagonal. Because our answer is in the form $(a\sqrt{b})(Φ)$ , $a$ and $b$ both equal $3$ , so $3+3=6$ . As a result, the final answer is $6$ .