To determine which measurement will require the use of irrational numbers, let's consider each of the options given for a cube with edge length \( a \) (a rational number).
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The diagonal of one of the cube’s faces: The formula for the diagonal \( d \) of a face of a cube, which is a square with edge length \( a \), is given by: \[ d = a\sqrt{2} \] Since \( \sqrt{2} \) is an irrational number, the diagonal of the face will be irrational if \( a \) is rational.
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The surface area of the cube: The surface area \( S \) of a cube is given by: \[ S = 6a^2 \] Since \( a^2 \) is rational (the square of a rational number is also rational), the surface area will also be rational.
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The volume of the cube: The volume \( V \) of a cube is given by: \[ V = a^3 \] Again, since \( a^3 \) is rational (the cube of a rational number is also rational), the volume will 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 given by: \[ A = a^2 \] Since \( a^2 \) is rational, the area of one face will be 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.
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