PSLE Gabriel 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 Gabriel continued building the steps in this way, what would be the height of the set of steps that had 312 cubes? Express your answer in cm.
Cubes needed for 4-cm steps = 4 x 3 = 4 x (1 + 2) = 12 cubes
Cubes needed for 6-cm steps = 4 x 6 = 4 x (1 + 2 + 3) = 24 cubes
Cubes needed for 8-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 = 312
Height x (Height + 1)2 = 312 ÷ 4
Height x (Height + 1)2 = 78
Height x (Height + 1) = 78 x 2
Height x (Height + 1) = 156
12 x 13 = 156
Number of cubes needed for the height of the set of steps that had 312 cubes = 12
Height of the set of steps that had 312 cubes
= 12 x 2
= 24 cm
Answer(s): 24 cm