The number of white and grey blocks in the quilts follows a sequence given by the Design number of the quilt
The numbers and patterns of the blocks are;
a. Patterns;
- The height of the white blocks is the Design number, while the rows are one more than the design number
- The height of the outer grey blocks is 2 blocks more than the Design number, while their rows are 3 blocks more than the Design number
b. The function is [tex]\underline{p(n) = n \cdot (n + 1)}[/tex]
c. The function is [tex]\underline{b(n) = (n + 2) \cdot (n + 3) - n \cdot (n + 1)}}[/tex]
d. [tex]\underline{t(n) = (n + 2) \cdot (n + 3) = b(n) - p(n) }[/tex]
e. The design chosen is Design 9
The number of blocks is 132 blocks
Reasons:
The pattern of the design are;
a. The column height of the grey blocks = 2 blocks more than the column height of the white blocks
Width of the outer grey blocks = 2 blocks more than the width of the inner white blocks
The number of white blocks in a design, n is the product of n and (n + 1)
The number of grey blocks in a design n, is the product of (n + 2) and (n + 3) less the number of white blocks in the design
b. Based on the above, the function that represent the number of picture blocks in Design n, is therefore;
c. Based on the above, the function that represent the number of border blocks in Design n, is therefore;
- b(n) = (n + 2)·(n + 3) - n·(n + 1)
d. The total number of blocks, t(n) = (n + 2)·(n + 3)
t(n) = (n + 2)·(n + 3) - n·(n + 1) + n·(n + 1)
e. The given number of picture blocks = 90
The number of picture blocks in a design, n = n × (n + 1)
Therefore, given that the number of picture blocks is 90, we have;
90 = n × (n + 1)
n² + n - 90 = 0
(n + 10)·(n - 9) = 0
n = 9, or n = -10
Therefore, the design museum chooses is Design 9
The total number of blocks in a quilt design, t(n) = (n + 2)·(n + 3)
Total blocks in the design the art museum choses, t(9), is given as follows;
t(9) = (9 + 2)·(9 + 3) = 132 blocks
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