To determine which measurement will require the use of irrational numbers when the edge length \( a \) of the cube is a rational number, let’s analyze each measurement:
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Surface area of the cube: The surface area \( S \) of a cube with edge length \( a \) is given by the formula: \[ S = 6a^2 \] Since \( a \) is rational, \( a^2 \) will also be rational, and thus \( S \) is rational.
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Volume of the cube: The volume \( V \) of a cube with edge length \( a \) is given by the formula: \[ V = a^3 \] Again, since \( a \) is rational, \( a^3 \) is rational, so \( V \) is rational.
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Area of one of the cube's faces: The area \( A \) of one face of the cube is: \[ A = a^2 \] As established earlier, \( a^2 \) is rational, therefore \( A \) is rational.
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Diagonal of one of the cube's faces: The diagonal \( d \) of one face of the cube can be found using the Pythagorean theorem: \[ d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \] Since \( \sqrt{2} \) is an irrational number, \( a\sqrt{2} \) will also be irrational for any rational \( a \neq 0 \).
Based on this analysis, the measurement that will require the use of irrational numbers is the diagonal of one of the cube's faces.
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