the manager of a grovery store asks a store clerk to rearrange a display of canned vegtables in a triangular pyramid. Assume all cans are thr same size and shape.
in the first pyramid their is 1 can in the base, second pyramid has 3 cans in the base and 3rd pyramid shown has 6 cans in the base.
A) if the clerk has 100 cans to use, how many of those can are needed to create the tallest complete pyramid shape?
B) how many can make up the bottom level of the pyramid in part A
c) how many levels is the pyramid in part A
For this question i have no idea where to start or if theirs a formula i can use. thank you for your help
To solve this problem, we can derive a formula for the number of cans needed for each level of the pyramid.
A) To find the number of cans needed to create the tallest complete pyramid shape with 100 cans, we need to determine the largest possible complete pyramid that can be formed with 100 cans. We can do this by finding the sum of the triangular numbers until their sum is less than or equal to 100.
The formula to calculate the sum of the first n triangular numbers is given by: sum = n(n+1)(n+2)/6
We can use this formula to calculate the sum of triangular numbers.
Let's solve for n:
n(n+1)(n+2)/6 ≤ 100
n(n+1)(n+2) ≤ 600
Testing values of n, we find that n = 8 is the largest value that satisfies the inequality. This means the largest complete pyramid that can be formed with 100 cans has a height of 8 levels.
Now, to find the number of cans needed for this pyramid, we calculate the sum of the first 8 triangular numbers:
sum = 8(8+1)(8+2)/6 = 8(9)(10)/6 = 8(3)(5) = 120 cans
Therefore, to create the tallest complete pyramid shape with 100 cans, you would need 120 cans.
B) In part A, we found that there are 120 cans in the tallest complete pyramid shape. The bottom level of the pyramid is the largest triangular number that is less than or equal to 120. We can find this by reverse calculating the triangular number.
Let's solve for n:
n(n+1)/2 ≤ 120
n(n+1) ≤ 240
Testing values of n, we find that n = 15 is the largest value that satisfies the inequality. This means the bottom level of the pyramid in part A consists of 15 cans.
C) In part A, we found that the tallest complete pyramid shape with 100 cans has a height of 8 levels. Therefore, the pyramid in part A consists of 8 levels.
To solve this problem, we can use some basic mathematical patterns and formulas. Let's break down each part of the question and find the answers step by step:
A) To find the number of cans needed to create the tallest complete pyramid shape, we need to determine the maximum number of cans required to form a pyramid.
We know that the first pyramid has a base with 1 can, the second pyramid has a base with 3 cans, and the third pyramid has a base with 6 cans. By observing this pattern, we can see that the number of cans required for each level increases by consecutive odd numbers (1, 3, 5, 7, ...).
Therefore, we can find the number of cans needed by adding consecutive odd numbers until we reach or exceed 100 cans.
The sequence of consecutive odd numbers we need to add is 1, 3, 5, 7, 9, 11, ... until we reach or exceed 100.
Let's calculate the sum of these numbers:
1 + 3 + 5 + 7 + 9 + 11 + ...
To calculate this sum, we can use the formula for the sum of consecutive odd numbers:
Sum = (n^2)
where n is the number of terms in the sequence.
In our case, we want the sum to be at least 100, so we need to solve the equation:
100 ≤ n^2
Simplifying the equation:
n^2 ≥ 100
Taking the square root of both sides:
n ≥ √100
n ≥ 10
So, we need at least 10 terms in our sequence of consecutive odd numbers to exceed 100. This means we would need the sum of the first 10 odd numbers.
Now, let's calculate the sum of the first 10 odd numbers:
Sum = (10^2) = 100
Therefore, we need at least 100 cans to create the tallest complete pyramid shape.
B) In part A, the tallest complete pyramid shape required 100 cans. To find the number of cans forming the bottom level, we can calculate the square of the number of cans needed for the tallest complete pyramid shape.
So, the number of cans in the bottom level will be the square of 10, which is 100.
Therefore, 100 cans make up the bottom level of the pyramid in part A.
C) To determine the number of levels in the pyramid, we need to find the number of terms in our sequence of consecutive odd numbers, which is 10.
Therefore, the pyramid in part A has 10 levels.
I hope this explanation helps! Let me know if you have any further questions.
the number of cans on each layer is a triangular number.
So, you need the sum of such numbers.
you could start here:
https://en.wikipedia.org/wiki/Tetrahedral_number