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Shapes in the form of small equilateral triangles can be made using matchsticks of equal length - Leaving Cert Mathematics - Question 9 - 2013
Question 9
Shapes in the form of small equilateral triangles can be made using matchsticks of equal length. These shapes can be put together into patterns. The beginning of a s... show full transcript
Worked Solution & Example Answer:Shapes in the form of small equilateral triangles can be made using matchsticks of equal length - Leaving Cert Mathematics - Question 9 - 2013
Step 1
Step 2
Step 3
Step 4
Find an expression, in n, for the number of matchsticks needed to turn the (n−1)th pattern into the nth pattern.
Answer
The expression for the number of matchsticks needed to transition from the (n-1)th pattern to the nth pattern is:
This indicates a linear relation between the pattern number and the matchsticks needed.
Step 5
The number of matchsticks in the nth pattern can be represented by the function un = an^2 + bn where a, b ∈ Q and n ∈ N. Find the value of a and the value of b.
Answer
After analyzing the pattern, we establish:
- From the second differences of matchsticks:
ightarrow a = \frac{3}{2}$$
- Plugging values back, we find:
Thus, we have determined:
Step 6
One of the patterns in the sequence has 4134 matchsticks. How many small triangles are in that pattern?
Answer
To find how many small triangles are in the pattern with 4134 matchsticks, we solve:
which leads to:
Now using the expression for small triangles:
Hence, the number of small triangles in that pattern is:
1,904,684 triangles.