Infinite Geometric Series

[F19]

Any help with this problem please! I have no idea where to start. Please include some work.

Thanks!

 

[fasteddie65]

On the first trip from B to AC, it travels a distance of 5?3.
Heading back to AB, it travels 5, then 2.5?3, then 2.5, etc.

(5 + 2.5 + 1.25 + ...) + (5 + 2.5 + 1.25 + ...)?3 =

Do you see what is needed here?

 

[F19]

On the first trip from B to AC, it travels a distance of 5?3.
Heading back to AB, it travels 5, then 2.5?3, then 2.5, etc.

(5 + 2.5 + 1.25 + ...) + (5 + 2.5 + 1.25 + ...)?3 =

Do you see what is needed here?

I understand how you got 5 root 3, but not how you got 5 traveling back to AB. Could you please clarify?

 

[mmm4444bot]

… how you got 5 traveling back to AB. Could you please clarify?

I don't get 5; I get 5 * sqrt(3) * sqrt(3)/2.

Do you realize? All of the triangles in the diagram are 30-60-90 triangles.

So, we can calculate each segment of the bug path in terms of sine.

First segment: 5 * sqrt(3)

Second segment: 5 * sqrt(3) * sqrt(3)/2

Third segment: 5 * sqrt(3) * sqrt(3)/2 * sqrt(3)/2

Fourth segment: 5 * sqrt(3) * sqrt(3)/2 * sqrt(3)/2 * sqrt(3)/2

Are these the values that you get?