Ring of Gold

First, we can substitue 1 for B in our side length proportion:

$\frac{A}{1}$ = $\frac{1}{C}$

then A = $\frac{1}{C}$ = 1 + C.

Solving for C gives us:

1 + C = $\frac{1}{C}$

C^2 + C = 1

C^2 + C - 1 = 0

Using Quadratic Formula, the positive solution is:

C = $\frac{sqrt(5) - 1}{2}$ .

That means that side A is given by:

A = 1 + $\frac{sqrt(5) - 1}{2}$

A = $\frac{sqrt(5) + 1}{2}$

Also known as the Golden Proportion.

The radius of the circular mold will be equal to half of the diagonal of the square with side A, so, using good old Pythagorean:

Diagonal 2r = sqrt[2*( $\frac{sqrt(5) + 1}{2}$ )^2]

2r = sqrt[2* $\frac{(sqrt(5) + 1)^2}{4}$ ]

2r = sqrt[ $\frac{(sqrt(5) + 1)^2}{2}$ ]

2r = $\frac{sqrt(5) + 1}{sqrt(2)}$

r = $\frac{sqrt(5) + 1}{2sqrt(2)}$ = $\frac{sqrt(5) + 1}{sqrt(8)}$ .

Since the depth of the mold is C cm, and since the inside of the ring is cylindical, the volume of the opening is given by:

V = pi C r^2

V = pi $\frac{sqrt(5) - 1}{2}$ $\frac{(sqrt(5) + 1)^2}{8}$

V = pi* $\frac{4sqrt(5) + 4}{16}$

V = pi* $\frac{sqrt(5) + 1}{4}$

or approximately 2.5416 cubic cm.