PSLE Reggie 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 Reggie continued building the steps in this way, what would be the height of the set of steps that had 165 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 = 165
Height x (Height + 1)2 = 165 ÷ 3
Height x (Height + 1)2 = 55
Height x (Height + 1) = 55 x 2
Height x (Height + 1) = 110
10 x 11 = 110
Number of cubes needed for the height of the set of steps that had 165 cubes = 10
Height of the set of steps that had 165 cubes
= 10 x 3
= 30 cm
Answer(s): 30 cm