PSLE Pierre 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 Pierre continued building the steps in this way, what would be the height of the set of steps that had 420 cubes? Express your answer in cm.
Cubes needed for 6-cm steps = 4 x 3 = 4 x (1 + 2) = 12 cubes
Cubes needed for 9-cm steps = 4 x 6 = 4 x (1 + 2 + 3) = 24 cubes
Cubes needed for 12-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 = 420
Height x (Height + 1)2 = 420 ÷ 4
Height x (Height + 1)2 = 105
Height x (Height + 1) = 105 x 2
Height x (Height + 1) = 210
14 x 15 = 210
Number of cubes needed for the height of the set of steps that had 420 cubes = 14
Height of the set of steps that had 420 cubes
= 14 x 3
= 42 cm
Answer(s): 42 cm