PSLE Jeremy wanted to build a set of steps with 3-cm cubes. The figures show how he built the steps from a height of 6 cm to 9 cm to 12 cm. If Jeremy continued building the steps in this way, what would be the height of the set of steps that had 108 cubes? Express your answer in cm.
Cubes needed for 6-cm steps = 3 x 3 = 3 x (1 + 2) = 9 cubes
Cubes needed for 9-cm steps = 3 x 6 = 3 x (1 + 2 + 3) = 18 cubes
Cubes needed for 12-cm steps = 3 x 10 = 3 x (1 + 2 + 3 + 4) = 30 cubes
Formula:
Number of cubes needed for the set of steps
= 3 x (1 + 2 + 3 + ... + Height)
= 3 x
(Height) x (Height + 1)2
3 x (Height) x (Height + 1)2 = 108
Height x (Height + 1)2 = 108 ÷ 3
Height x (Height + 1)2 = 36
Height x (Height + 1) = 36 x 2
Height x (Height + 1) = 72
8 x 9 = 72
Number of cubes needed for the height of the set of steps that had 108 cubes = 8
Height of the set of steps that had 108 cubes
= 8 x 3
= 24 cm
Answer(s): 24 cm