Solution 1 — Approximate Solution

To find the amount of framing, we need the side length of the octagon. 

For the right isosceles triangles, let \(x\) represent the length of the legs in cm and \(h\) the length of the hypotenuse and the side length of the frame.

By the Pythagorean Theorem,

Since \(h \gt\ 0\), \(h=\sqrt{2}\sqrt{x^2}\approx1.414x\). 

Solution 1 — Approximate Solution Continued

The width of the mirror is \(99\) cm.

The side length of the frame is \(h\approx1.414(29) \approx 41\) cm or ​​​​​​
\(0.41\) m.

The cost of the frame is approximately \((8)(0.41)(12.95) \approx$42.48\).

Solution 1 — A More Exact Solution

To find the amount of framing, we need the side length of the octagon. 

For the right isosceles triangles, let \(x\) represent the length of the legs in cm and \(h\) the length of the hypotenuse and the side length of the frame.

By the Pythagorean Theorem,

Since \(h \gt\ 0\), \(h=\sqrt{2}\sqrt{x^2}=\sqrt{2}x\).

Solution 1 — A More Exact Solution Continued

The width of the frame is \(99\) cm.

The side length of the frame is \(h=\sqrt{2}\left(\dfrac{99}{2+\sqrt{2}}\right) \approx 41\) cm or \(0.41\) m.

The cost of the frame is approximately \($42.48\).