# a theory about moth navigation is that moths fly at a constant angle relative to a light source.

a particular moth flies at 50 degrees to the light source,L, and it measures its bearing 10 times in one revolution.
Hint:one revolution is 360 degrees. 10 times makes the angle R1 L R0 equal to 360/10 (R1 here means the distance between L and the starting point of the moth. R2 is the distance between L and the first time the moth measures its bearing.)

1.)what happens if the number of times the moth measures its bearing increases to a very big number??

2)what does the limiting value appear to be?

3) what will the flight path be for x-->infinitive?

i drew a diagram for question 1 and figured out that as the number of times the moth measures its bearing increases to a very big number, the circle will get smaller and the moth will fly to the light sooner.but i can't figure out a formula for this...

i do not get question number 2.
question 3 is very similar to question number 2 right?? but i am still a little bit confused.=(

i have also worked out a formula for R1 in terms of R0:
R1=R0 times sin50/sin94)

so R(n) for all path is:
(let the angle the moth flies to the light source be A, and let times measured in one rev be B)

R(n)=sinA/sin{(360/B) +A}

i could try out question 1 and 3 by changing the value of B in that formula, but i don't know how to make a formula for question 1 and 3 by doing so...

I am really sorry if this looked very messy...but i will be so happy if some one can help me!thanks!! =)

## 1) If the number of times the moth measures its bearing increases to a very big number, the circle will become smaller and the moth will fly closer to the light source. This is because as the number of measurements increases, the angle between each measurement decreases, resulting in a tighter spiral towards the light.

2) The limiting value refers to the value that the measurements approach as the number of times the moth measures its bearing approaches infinity. In this case, the limiting value appears to be zero, as the moth gets closer and closer to the light source as more measurements are taken.

3) The flight path as x approaches infinity will be a spiral that converges towards the light source. As more and more measurements are taken, the spiral becomes tighter and the moth gets closer to the light source.

Regarding the formula you've worked out, R1 = R0 * sin(A) / sin((360/B) + A), it represents the distance between the starting point of the moth (R1) and the first time the moth measures its bearing (R2), in terms of the angle of flight to the light source (A) and the number of measurements in one revolution (B). This formula can be used to calculate the distance at each measurement point (R(n)) by substituting the appropriate values of A and B.

To summarize, as the number of measurements increases, the circle becomes smaller, the moth gets closer to the light source, and the flight path forms a converging spiral. The limiting value is zero, indicating that the moth will eventually reach the light source.