To determine which measurement will require the use of irrational numbers for a cube with edge length \( a \) (a rational number), let's analyze each option:
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The area of one of the cube’s faces: The area of a face of the cube is given by \( A = a^2 \). Since \( a \) is rational, \( a^2 \) is also rational.
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The surface area of the cube: The surface area \( S \) of a cube is calculated as \( S = 6a^2 \). Again, since \( a \) is rational, \( a^2 \) is rational, and thus \( 6a^2 \) is also rational.
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The diagonal of one of the cube’s faces: The diagonal \( d \) of one of the cube's faces 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, \( a\sqrt{2} \) is irrational (as it is the product of a rational number \( a \) and an irrational number \( \sqrt{2} \)).
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The volume of the cube: The volume \( V \) of the cube is calculated as \( V = a^3 \). Since \( a \) is rational, \( a^3 \) is also rational.
Based on this analysis, the measurement that will require the use of irrational numbers is:
The diagonal of one of the cube’s faces.