To analyze which measurement will require the use of irrational numbers, let's examine each of the options given the edge length \( a \), where \( a \) is a rational number:
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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 \( a \) is a rational number, \( a^2 \) is also rational, and thus \( S \) will be rational.
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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: \[ d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \] Since \( \sqrt{2} \) is an irrational number, the diagonal \( d \) will be irrational when multiplied by the rational number \( a \).
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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, \( a^3 \) will also be rational.
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Area of One of the Cube's Faces: The area \( A \) of one face is given by: \[ A = a^2 \] As \( a \) is rational, \( A \) will also be rational.
Conclusion:
Out of the options given, the measurement that will require the use of irrational numbers is the diagonal of one of the cube's faces.
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