To determine which measurement will require the use of irrational numbers for a cube with a rational edge length, let's analyze each option:
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The diagonal of one of the cube's faces: The diagonal \(d\) of a face of a cube with edge length \(a\) can be calculated using the Pythagorean theorem: \[ d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \] Since \(a\) is rational, \(a\sqrt{2}\) will be irrational because \(\sqrt{2}\) is an irrational number.
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The volume of the cube: The volume \(V\) of a cube with edge length \(a\) is given by: \[ V = a^3 \] Since \(a\) is rational, \(a^3\) is also rational.
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The surface area of the cube: The surface area \(S\) of a cube is calculated as: \[ S = 6a^2 \] Since \(a\) is rational, \(6a^2\) is also rational.
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The area of one of the cube's faces: The area \(A\) of one face is: \[ A = a^2 \] Since \(a\) is rational, \(a^2\) is also rational.
Based on this analysis, the only measurement that requires the use of irrational numbers is:
The diagonal of one of the cube's faces.
1 answer