PSLEAndy used 3-cm rods to build some structures. The first five structures are shown.
The table shows the number of rods used for each structure and the height of each structure.
Structure number |
Number of rods |
Height of the structure (cm) |
1 |
12 |
3 |
2 |
20 |
3 |
3 |
28 |
6 |
4 |
33 |
6 |
5 |
41 |
9 |
... |
... |
... |
16 |
a |
b |
- Find the value of a.
- Find the value of b.
- What was the height of Structure 139?
- How many rods were used to build Structure 139?
(a)
Structure 1:
Rods used = 12
Structure 2:
Rods used
= 12 + 8
= 20
Structure 3:
Rods used
= 12 + 8 + 8
= 20 + 8
= 28
Structure 4:
Rods used
= 12 + 8 + 8 + 5
= 20 + 13
= 33
Structure 5:
Rods used
= 12 + 8 + 8 + 5 + 8
= 20 + 13 + 8
= 41
Structure 6: Rods used
= 12 + 8 +
8 + 5 +
8 + 5 = 20 + 13 + 13
= 46
Number of 13s to be added to 20 to obtain the number of rods used for Structure 16
= (Structure number - 2) ÷ 2
= (16 - 2) ÷ 2
= 14 ÷ 2
= 7
Number of rods to be used for Structure 16
= 20 + 13 x 7
= 111
(b)
Height for even numbered structure = (Figure number ÷ 2) x length of each rod
Height for Structure 16
= (16 ÷ 2) x 3
= 8 x 3
= 24 cm
(c)
Height of structure
Structure 1: 3 cm = 1 x 3
Structure 2: 3 cm = 1 x 3
Structure 3: 6 cm = 2 x 3
Structure 4: 6 cm = 2 x 3
Structure 5: 9 cm = 3 x 3
Formula for odd number structure:
Height of structure =
(Structure number + 1)2 x 3
Height of structure
=
(139 + 1)2 x 3
= 140 ÷ 2 x 3
= 70 x 3
= 210 cm
(d)
Pattern for number of rods for odd numbered structure
Structure 3: 28
Structure 5: 28 + (13 x 1)
Structure 7: 28 + (13 x 2)
Structure 9: 28 + (13 x 3)
Number of 13s to be added to 28 to obtain the number of rods used for Structure 139
= (Structure number - 3) ÷ 2
=
(139 - 3)2 = 68
Number of rods used to build Structure 139
= 28 + 68 x 13
= 912
Answer(s): (a) 111; (b) 24 cm; (c) 210 cm; (d) 912