To identify which measurement will require the use of irrational numbers, we can analyze the measurements associated with a cube with edge length \( a \), a rational number.
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The diagonal of one of the cube’s faces: The diagonal \( d \) of a 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, the diagonal will be irrational if \( a \) is rational.
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The area of one of the cube’s faces: The area \( A \) of a face of the cube is: \[ A = a^2 \] This is rational as the square of a rational number is rational.
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The volume of the cube: The volume \( V \) of the cube is: \[ V = a^3 \] This is rational since the cube of a rational number is rational.
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The surface area of the cube: The surface area \( S \) of the cube is: \[ S = 6a^2 \] This is also rational as it involves the square of a rational number.
Therefore, the measurement that will require the use of irrational numbers is the diagonal of one of the cube’s faces.
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