Mia is stacking copies of a new book in a square-pyramid display by the front window of her bookstore. for each consecutive layer, she places one book where four meet.
A. I the bottom row of the display has 144 books and there is one book on the top, determine how many rows of books are in her pyramid.
B. Explain how to find the total number of books in the display.
C. If she removes the top four rows, how many books are left in the pyramid display?
You might want to make a sketch to show that the bottom level is a 12 by 12 square
the second level is a 11 by 11 square, etc
So your sum is 12^2 + 11^2 + 10^2 + ... 2^2 + 1^2
you could just add up these numbers to get 650
or you could use a formula that says
the sum of squares of consecutive numbers from 1 to n is
n(n+1)(2n+1)/6 which gives us 650
for the last part, find the sum of books removed at the top and subtract that from 650
btw, did you realize that the books would have to be square in shape?
A. To determine the number of rows in Mia's pyramid, we need to find the number of layers or steps in the pyramid. Since each layer consists of one less book than the layer below it, we can start by finding the largest triangular number that is less than 144.
A triangular number is the sum of the natural numbers starting from 1. The formula to calculate the nth triangular number is:
T(n) = n(n+1)/2
If we solve the equation T(n) = 144, we can find the largest triangular number below 144. We can set up the equation as follows:
n(n+1)/2 = 144
n(n+1) = 288
n^2 + n - 288 = 0
By factoring or using the quadratic formula, we find that n = 16 is the largest integer value that satisfies the equation.
Therefore, Mia's pyramid has 16 rows of books.
B. To find the total number of books in the display, we need to sum the number of books in each layer. We know that the bottom row has 144 books, and each subsequent layer has one less book than the layer below it.
We can use the formula for the sum of an arithmetic series to calculate the total number of books:
S = (n/2)(2a + (n-1)d)
Where:
S = Sum of the series
n = Number of terms (rows in this case)
a = First term (number of books in the bottom row)
d = Common difference (change in the number of books between consecutive rows)
In this case, n = 16, a = 144, and d = -4 (since each row has 4 books less than the previous row).
Plugging in the values, we get:
S = (16/2)(2(144) + (16-1)(-4))
S = 8(288 - 60)
S = 8(228)
S = 1824
Therefore, the total number of books in the display is 1824.
C. If Mia removes the top four rows, we need to calculate how many books are left in the pyramid display.
Since each layer has one less book than the layer below it, the top four rows contain a total of 4 + 3 + 2 + 1 = 10 books.
Subtracting 10 from the previous total of 1824, we find that there are 1814 books left in the pyramid display.