a store does a display of 200 balls, he places them in the shape of a square pyramid. the bottom layer has 64 balls for each consecutive layer of balls, one ball is placed where 4 balls meet. how may layers will be in the pyramid? how many will be left over?
To figure out how many layers there will be in the pyramid, we need to determine the maximum number of balls each layer can accommodate.
In a square pyramid, the number of balls in each layer follows a pattern. The bottom layer has 64 balls, and each consecutive layer has 4 fewer balls than the layer below it.
So, to find out the maximum number of layers, we need to calculate how many times we can subtract 4 from 64 until we reach 0.
64 - 4 = 60
60 - 4 = 56
56 - 4 = 52
52 - 4 = 48
48 - 4 = 44
44 - 4 = 40
40 - 4 = 36
36 - 4 = 32
32 - 4 = 28
28 - 4 = 24
24 - 4 = 20
20 - 4 = 16
16 - 4 = 12
12 - 4 = 8
8 - 4 = 4
4 - 4 = 0
We see that we can subtract 4 a total of 16 times until we reach 0. Therefore, there will be 16 layers in the pyramid.
To calculate the remaining balls, we need to add up the number of balls in each layer. We can use the sum of an arithmetic series formula:
S = (n/2)(2a + (n - 1)d),
where S is the sum of the series, n is the number of terms, a is the first term, and d is the common difference.
In our case, the first term 'a' is 64, the common difference 'd' is -4 (since each layer has 4 fewer balls), and the number of terms 'n' is 16 (the number of layers).
S = (16/2)(2*64 + (16 - 1)*(-4))
S = 8(128 + 15*(-4))
S = 8(128 - 60)
S = 8(68)
S = 544
So, there will be a total of 544 balls in the pyramid. To calculate the remaining balls, we subtract this number from the initial 200 balls.
Remaining balls = 200 - 544 = -344
It seems that there will be a deficit of 344 balls.