To determine which measurement will require the use of irrational numbers when the edge length \( a \) of a cube is a rational number, we can analyze each of the options provided:
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Volume of the cube: The volume \( V \) of a cube with edge length \( a \) is given by: \[ V = a^3 \] Since \( a \) is rational, \( a^3 \) will also be rational.
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Diagonal of one of the cube’s faces: The diagonal \( d \) of a face of the cube (which is a square) 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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Surface area of the cube: The surface area \( S \) of a cube is given by: \[ S = 6a^2 \] Since \( a \) is rational, \( 6a^2 \) will also be rational.
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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, \( a^2 \) will also be rational.
From this analysis, the only measurement that requires the use of irrational numbers is the diagonal of one of the cube’s faces, as it involves \( a\sqrt{2} \) which is irrational.
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