To determine which measurement will require the use of irrational numbers, let's analyze each option in the context of a cube with an edge length \( a \) (a rational number).
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Volume of the cube: The volume \( V \) of a cube is given by the formula \( V = a^3 \). Since \( a \) is rational, \( a^3 \) is also rational. Thus, the volume requires only rational numbers.
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Diagonal of one of the cube’s faces: The diagonal \( d \) of a face of the cube can be calculated using the Pythagorean theorem. For a face with both edges of length \( a \), the diagonal is given by \( d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \). Since \( \sqrt{2} \) is an irrational number, the diagonal involves irrational numbers.
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Surface area of the cube: The surface area \( S \) of a cube is given by the formula \( S = 6a^2 \). Since \( a^2 \) is rational and multiplying by a rational number (6) keeps it rational, the surface area only involves rational numbers.
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Area of one of the cube’s faces: The area \( A \) of one face is given by \( A = a^2 \). Since \( a^2 \) is also rational (as \( a \) is rational), the area of one face consists only of rational numbers.
Given this analysis, the only measurement that requires the use of irrational numbers is:
the diagonal of one of the cube’s faces.