PSLE Cole 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 Cole continued building the steps in this way, what would be the height of the set of steps that had 84 cubes? Express your answer in cm.
Cubes needed for 8-cm steps = 3 x 3 = 3 x (1 + 2) = 9 cubes
Cubes needed for 12-cm steps = 3 x 6 = 3 x (1 + 2 + 3) = 18 cubes
Cubes needed for 16-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 = 84
Height x (Height + 1)2 = 84 ÷ 3
Height x (Height + 1)2 = 28
Height x (Height + 1) = 28 x 2
Height x (Height + 1) = 56
7 x 8 = 56
Number of cubes needed for the height of the set of steps that had 84 cubes = 7
Height of the set of steps that had 84 cubes
= 7 x 4
= 28 cm
Answer(s): 28 cm