PSLE Paul wanted to build a set of steps with 4-cm cubes. The figures show how he built the steps from a height of 8 cm to 12 cm to 16 cm. If Paul continued building the steps in this way, what would be the height of the set of steps that had 144 cubes? Express your answer in cm.
Cubes needed for 8-cm steps = 4 x 3 = 4 x (1 + 2) = 12 cubes
Cubes needed for 12-cm steps = 4 x 6 = 4 x (1 + 2 + 3) = 24 cubes
Cubes needed for 16-cm steps = 4 x 10 = 4 x (1 + 2 + 3 + 4) = 40 cubes
Formula:
Number of cubes needed for the set of steps
= 4 x (1 + 2 + 3 + ... + Height)
= 4 x
(Height) x (Height + 1)2
4 x (Height) x (Height + 1)2 = 144
Height x (Height + 1)2 = 144 ÷ 4
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 144 cubes = 8
Height of the set of steps that had 144 cubes
= 8 x 4
= 32 cm
Answer(s): 32 cm