Height of one small triangle = [tex]\frac{\sqrt{3} }{2}(a)[/tex]
[tex]=\frac{\sqrt{3} }{2} (2.48)[/tex]
[tex]= (1.732) (1.24)[/tex]
= 2.148 inches
So, height of each row = height of each small triangle = 2.148 inches.
Also, base length of each small triangle = length of the toothpick = 2.48 inches.
So, if we divide the room width by length of one toothpick, we can find the number of small triangles in the bottom row.
Note that I converted the room width to inches.
Number of triangles in the bottom row = [tex]\frac{22(12)}{2.48}[/tex]
[tex]= \frac{264}{2.48}[/tex]
= 106 (rounded)
Also, if we look at the pattern, we can find there is one triangle in the top row, two triangles are in the second row, three triangles are in the third row and so on.
So, in the bottom row there are 106 triangles and it means that there are 106 rows.
Since, height of one triangle is 2.148 inches and there are 106 rows, lets check whether the height of big triangle fits within room's height.
Multiply 106 and 2.148.
106 × 2.148 = 227 approximately.
Room's height = 28 ft = 28(12) inches = 336 inches.
Since 227 < 336,
the big triangle would exactly fit in the room.
Now, lets calculate the total number of small triangles.
According to the pattern,
total number of triangles = 1 + 2 + 3 + ....+ 106
[tex]=\frac{106(107)}{2}[/tex]
= 53 × 107
= 5671
Hence, there are 5671 small triangles and since a small triangle needs 3 toothpicks,
total number of toothpicks = 3 × 5671 = 17013.
Note that all small triangles are equilateral.
