To determine which measurement will require the use of irrational numbers, let's analyze each option:
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The volume of the cube: The volume \( V \) of a cube with edge length \( a \) is calculated using the formula \( V = a^3 \). Since \( a \) is a rational number, the volume will also be a rational number.
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The diagonal of one of the cube’s faces: The diagonal \( d \) of one of the cube's faces (which is a square) can be found using the Pythagorean theorem. If the edge length is \( a \), 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 will be irrational if \( a \) is not zero.
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The surface area of the cube: The surface area \( S \) of a cube is calculated using the formula \( S = 6a^2 \). As long as \( a \) is rational, the surface area will also be rational.
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The area of one of the cube’s faces: The area \( A \) of one face of the cube is \( A = a^2 \). Since \( a \) is rational, the area will be rational.
Based on this analysis, the measurement that requires the use of irrational numbers is:
The diagonal of one of the cube’s faces.