in a following pattern, one hexagon takes 5 toothpicks to build, two hexagons take 11 toothpicks to build, and so on. How many toothpicks would it take to build (a) 10 hexagons? (b) n hexagons?
a 50
b ???
I am not yet certain of the pattern, Frankly, I would have expected
one hexagon takes 6
two hexagon takes 11
three take 16
and n would take 6+(n-1)5
10 hexagons would take 51
In don't know why you did not put the reat
To find the pattern and determine the number of toothpicks required to build a certain number of hexagons, we can analyze the given information.
Given that one hexagon takes 5 toothpicks to build and two hexagons take 11 toothpicks to build, we can find the number of additional toothpicks required to add each new hexagon.
To find the additional toothpicks required to add each new hexagon, subtract the toothpicks required for the previous situation from the toothpicks required for the current situation.
Number of hexagons: 1 2 3 4 5 6 7 8 9 10 ...
Toothpicks required: 5 11 17 23 29 35 41 47 53 ??? ...
From the given information, we can see that when one hexagon is added, the number of additional toothpicks required is 11 - 5 = 6.
To find the answer to each question:
(a) To find the number of toothpicks required for 10 hexagons, we can apply the pattern:
Toothpicks required = (Number of hexagons - 1) * 6 + 5
Toothpicks required = (10 - 1) * 6 + 5
Toothpicks required = 9 * 6 + 5
Toothpicks required = 54 + 5
Toothpicks required = 59
Therefore, it would take 59 toothpicks to build 10 hexagons.
(b) For 'n' hexagons, the formula can be generalized as follows:
Toothpicks required = (Number of hexagons - 1) * 6 + 5
Toothpicks required = (n - 1) * 6 + 5
So for 'n' hexagons, it would take (n - 1) * 6 + 5 toothpicks to build.
To find the total number of toothpicks needed to build a certain number of hexagons, we need to identify the pattern and then use that pattern to calculate the required number of toothpicks.
Let's first examine the pattern:
Number of Hexagons: 1 2 3 4 5 ...
Number of Toothpicks: 5 11 ?? ?? ?? ...
To find the pattern, let's compare the number of hexagons and the number of toothpicks.
- When there is 1 hexagon, we need 5 toothpicks.
- When there are 2 hexagons, we need 11 toothpicks.
- By examining the difference between the number of toothpicks for each consecutive number of hexagons, we can see that the difference increases by 6 (11 - 5 = 6).
Using this pattern, we can deduce that for each additional hexagon, we will need 6 more toothpicks.
Now, let's calculate the total number of toothpicks needed for different scenarios:
(a) To find the number of toothpicks required for 10 hexagons:
We can start with the number of toothpicks needed for 1 hexagon (which is 5) and add 6 toothpicks for each additional hexagon. So, to find the total number of toothpicks for 10 hexagons, we can use the formula:
Total number of toothpicks = (Number of hexagons - 1) * 6 + Number of toothpicks for 1 hexagon
Total number of toothpicks = (10 - 1) * 6 + 5
Total number of toothpicks = 9 * 6 + 5
Total number of toothpicks = 54 + 5
Total number of toothpicks = 59
Therefore, it would take 59 toothpicks to build 10 hexagons.
(b) To find the general formula for finding the total number of toothpicks needed for any number of hexagons (n):
We can use the same formula as above:
Total number of toothpicks = (Number of hexagons - 1) * 6 + Number of toothpicks for 1 hexagon
Total number of toothpicks = (n - 1) * 6 + 5
So, the general formula for the total number of toothpicks required to build n hexagons is:
Total number of toothpicks = (n - 1) * 6 + 5
Therefore, for any value of n, you can substitute it into the formula to calculate the number of toothpicks needed to build n hexagons.