PSLE Valen wanted to build a set of steps with 2-cm cubes. The figures show how he built the steps from a height of 4 cm to 6 cm to 8 cm. If Valen continued building the steps in this way, what would be the height of the set of steps that had 198 cubes? Express your answer in cm.
Cubes needed for 4-cm steps = 3 x 3 = 3 x (1 + 2) = 9 cubes
Cubes needed for 6-cm steps = 3 x 6 = 3 x (1 + 2 + 3) = 18 cubes
Cubes needed for 8-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 = 198
Height x (Height + 1)2 = 198 ÷ 3
Height x (Height + 1)2 = 66
Height x (Height + 1) = 66 x 2
Height x (Height + 1) = 132
11 x 12 = 132
Number of cubes needed for the height of the set of steps that had 198 cubes = 11
Height of the set of steps that had 198 cubes
= 11 x 2
= 22 cm
Answer(s): 22 cm