a playground is being designed where children can interact with their friends in certain combinations. If there is one child, there can be zero interactions. If there are two children, there can only be three interactions. If there are three children, there can only be nine interactions. If there are four children there can be 18 interactions. How many interactions will there be for nine children?
So I am looking at ordered pairs: (1,0), (2,3), (3,9), (4,18) , all of the form (x,y)
the second differences of the y values is constant, suggesting a quadratic relationship.
let y = ax^2 + bx + c
at (1,0) ---> a + b + c = 0 ,#1
at (2,3) ---> 4a + 2b + c = 3 , #2
at (3,9) ---> 9a + 3b + c = 9 , #3
subtract #1 from #2 ----> 3a + b = 3
subtract #2 from #3 ----> 5a + b = 6
now subtract those two:
2a = 3
a = 3/2
sub into 3a + b = 3 to find b = -3/2
subbing those back into a+b+c=0 , we get c = 0
so y = (3/2)x^2 - (3/2)x = 3x(x - 1)/2
testing for the data value we did not use,
if x = 4, y = 12(3)/2 = 18 , ok then!!!
when x = 9 , (nine children)
y = 27(8)/2 = 108 <------ number of interactions for 9 children
To find the number of interactions for nine children, we need to analyze the pattern in the given information.
Let's look at the number of interactions for each number of children:
- One child: zero interactions
- Two children: three interactions
- Three children: nine interactions
- Four children: 18 interactions
We can observe that the number of interactions increases with each additional child. However, the rate at which it increases is not consistent.
From the given information, we can see that the number of interactions doubles as we increase the number of children. For example, going from two to three children results in a three-fold increase (from three to nine), and going from three to four children results in a two-fold increase (from nine to 18).
Since we know that the number of interactions doubles each time, we can calculate the number of interactions for nine children by doubling the previous number of interactions (which is 18 for four children).
Doubling 18, we get:
18 * 2 = 36
Therefore, there will be 36 interactions for nine children in the playground.