Trisect each side of an equilateral triangle of side length 2 and construct an equilateral triangle on the center third of each side. Omit the side of each new triangle that was once a part of the original triangle. Repeat the trisection, construction, and omission steps on each segment formed above, creating the figure below. When this process is repeated indefinitely, a new figure is approached. The square of the area of this new figure can be written as

where p and q are relatively prime positive integers. Find p + q. ps. the figure is a koch flake. pretty sure the answer is 223, but i some conformation. thank you :)

shmewy Jun 27, 2024

#1**+1 **

I think it's just a glitch but the problem is missing some values.

We need to know that the area of the square can be written as for p and q to solve this problem.

I think the latex didn't go through.

Can you please just fix the part for the equation?

I'll try to solve it then!

Thanks! ::)

NotThatSmart Jun 27, 2024

#2**0 **

Trisect each side of an equilateral triangle of side length 2 and construct an equilateral triangle on the center third of each side. Omit the side of each new triangle that was once a part of the original triangle. Repeat the trisection, construction, and omission steps on each segment formed above, creating the figure below. When this process is repeated indefinitely, a new figure is approached. The square of the area of this new figure can be written as p/q

where p and q are relatively prime positive integers. Find p + q. ps. the figure is a koch flake. pretty sure the answer is 223, but i some conformation. thank you :)

this is the edited version. thank u :)

shmewy Jul 2, 2024

#3**+1 **

This is a tough questipn! It takes some thought to solve! Good one!

Now, let's note something really important first.

The area of the koch flake can be written in the form

where a_0 was the original area and n was the iteration number.

The limit of the area in terms of sidelength s is

Thus, plugging in 2, we have

\(\frac{8\sqrt3}{5}\)

Now, we square it and see what we get. We have

\(\frac{64(3)}{25} = \frac{192}{25}\)

So \(p=192; q=25\)

Thus, \(p+q = 25+192 = 217\)

Hmmm, I'm not sure if I'm correct. Can you explain your thought process. I think I misunderstood my own logic...

I am pretty sure I'm wrong...ummm, is there a way you can check?

Thanks! :)

~NTS

NotThatSmart Jul 2, 2024