PSLE Ivan 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 Ivan 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 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 = 84
Height x (Height + 1)2 = 84 ÷ 4
Height x (Height + 1)2 = 21
Height x (Height + 1) = 21 x 2
Height x (Height + 1) = 42
6 x 7 = 42
Number of cubes needed for the height of the set of steps that had 84 cubes = 6
Height of the set of steps that had 84 cubes
= 6 x 2
= 12 cm
Answer(s): 12 cm