Question

This entire problem is AQ. A farmer plants apple trees in a square pattern. In order to protect the apple trees against the wind he plants conifer trees all around the orchard. 

Here is a diagram of the situation where you can see the pattern of apple trees and conifer trees for the first 4 cases: 

AQ: Question Set: 

Note: Be sure to use Polya's problem solving process and include parts a-d in steps 1-4. 

a. How do you see the pattern growing? Extend the pattern by drawing more cases. 

b. Suppose the farmer wants to make a much larger orchard with many rows of apple trees. As the farmer makes the orchard bigger, following this pattern, which will increase more quickly: the number of apple trees or the number of conifer trees? How do the growths compare with each other? Justify your answer using multiple representations: the visual pattern, equations, tables, written descriptions and graphs. 

c. Write a description of how you would determine the number of conifer and apple trees in any case number. 

d. Write general expressions so you can calculate the number of conifer and apple trees in any case number.

Answer 1

a) I am extending this pattern till n=6

n	No. of apple trees	No. of conifer trees
1	1	8
2	4	16
3	9	24
4	16	32
5	25	40
6	36	48

b) Check the below points for which tree will increase quickly-

• Apple trees = n * n and conifer trees =8 * n both formulas have a factor n, but apple trees have another n which will get larger where factor 8 stays the same. The number of apple trees increases more quickly.

• The number of apple trees increases faster because that number is being squared instead of multiplied by 8

• The number of apple trees is quadratic. The number of conifer trees is linear. So apple trees will increase faster.

• And with the help of Big Oh representation, you can understand like O(n*n) grows faster than O(8*n).

c) As from the above explanation, it's clear that apple trees are increasing in terms of n^2, and conifer trees are increasing with 8 * n where n >=0.

d) Number of apple trees = n²

Number of conifer trees= 8 * n where n ≥ 0.