Yc.

6. The plants in a certain garden are arranged in four concentric circles, i.e., circles with a common center. Plants are spaced at 1-foot intervals, and the radius of each circle is 10 feet greater than the next one inside it. The first circle has a 10-foot radius. In any given time interval, 20% of the bugs in one circle will move to the next circle out from the center, and 20% to the next circle in toward the center. Note that this refers to the total number of bugs in each circle, not on each plant. Each plant initially has 50 bugs.

The bugs on the innermost circle have no circle inside them so they don't move inward. 20% of the bugs on the outer circle get lost.

· After the first transition, how many bugs will there be on each of the first four circles, and how many bugs on each plant in each of those circles?

Write the total number of bugs on all the trees after each transition as a sequence and analyze as best you can the behavior of the sequence.

The difference between the bugs on trees and trees in circles is that the bugs are now in a group that is inside a group. It’s almost the same except you look at the problem differently. Instead of looking at the trees, you now look at a group of trees. You must also understand that there are more trees in the outer circle, than there is trees in the inner circle, so you must calculate how many trees are in a circle than multiply that by the number of bugs. You can solve this through many different calculations using percentages from one circle to another, or you can solve this problem through matrices that can take you through many transitions in few calculations.

The matrix equation for this problem is:

[.8 .2 0 0] * [50*31.4]

[.2 .6 .2 0] [50*62.8]

[0 .2 .6 .2] [50*94.2]

[0 0 .2 .6] [50*125.6]

I gave you the equation, you do the math.

Since there is a 20% shift to each neighboring tree, the first circle of trees only has 80% that stays since 20% of it moves out, and it also gets 20% of the of the second circle of trees. The second circle of trees has 60% that stays because 20% go to the first circle and 20% goes to the third circle, but it also gets 20% of the first circle and 20% of the third circle. The third step is exactly the same as the second step. But the last circle of trees loses 20% to unintelligent bugs who get lost and 20% to the third tree. So it gets 20% from the third tree and 60% of itself.