Michal has 42 identical cubical blocks, each one with an edge of 1 cm. From all of these blocks, he built a rectangular prism with a base perimeter equal to 18 cm. What is the height of the prism which he built?

the area of the base is a factor of 42

with a perimeter of 18, the possible block arrangements are __
8 X 1 __ area 8
7 X 2 __ area 14
6 X 3 __ area 18
5 X 4 __ area 20

14 is the only factor of 42, so the height must be 3

To find the height of the prism, we can use the formula V = lwh, where V is the volume of the prism, l is the length, w is the width, and h is the height.

Since the blocks are identical and each one has an edge of 1 cm, the length, width, and height of the prism will be a multiple of the edge length (1 cm).

Let's assume the length of the prism is L, the width is W, and the height is H (all in cm).

Given that the base perimeter is 18 cm, we can express the perimeter in terms of L and W:
Perimeter = 2L + 2W = 18.

Since all edges of the base of the prism are made up of cubical blocks, dividing the total length of the base (2L + 2W) by the edge length (1 cm) will give us the number of blocks used.

Number of blocks used = (2L + 2W) / 1.

Since Michal has 42 identical blocks available, we can set up an equation:
(2L + 2W) / 1 = 42.

Simplifying the equation:
2L + 2W = 42.

Now, we have a system of equations:
2L + 2W = 42 ----(1)
2L + 2W = 18 ----(2)

Since equations (1) and (2) are equivalent, we can determine that the height of the prism (H) is the same as the height of the cubical blocks, which is 1 cm.