1
To solve this problem, we need to determine the number of cuboids with distinct dimensions whose volume equals the volume of a cube with sides measuring \(2 \, \text{cm}\) (since \(2^3 = 8 \, \text{cm}^3\)).
Given: The volume of the cube is \(8 \, \text{cm}^3\). The formula to calculate this is:
\(V_{\text{cube}} = a^3 = 8\), where \(a\) is the side of the cube.
Solving for \(a\), we find:
\(a = \sqrt[3]{8} = 2\)
Thus, the side of the cube is \(2 \, \text{cm}\).
Now, consider the cuboid with dimensions \(x, y, z\), where the volume is given as:
\(V_{\text{cuboid}} = x \cdot y \cdot z = 8\)
We are given the condition \(x > y > z\) and that \(x, y, z\) are natural numbers. We will check every set of numbers that multiply to \(8\):
- If \(z = 1\), then \(x \cdot y = 8\). Checking integer pairs, possible solutions are:
- \(y = 2, x = 4\), leading to the triplet \((4, 2, 1)\). This satisfies \(x > y > z\).
- Any other combinations such as \(y = 4, x = 2\) don't satisfy the condition \(x > y > z\).
- If \(z = 2\), then \(x \cdot y = 4\). Checking integer pairs for \(y\) and \(x\) doesn't provide additional solutions since \({2, 1}\) already uses these factors and the only valid order found is \({4, 2, 1}\).
Hence, only one valid set of natural number dimensions satisfies all the conditions: \((4, 2, 1)\).
The number of distinct cuboids is 1.
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