To determine which measurement of the cube will require the use of irrational numbers, we can analyze each option based on the properties of a cube and the calculations involved.
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The volume of the cube: The volume \( V \) of a cube is calculated using the formula \( V = a^3 \). Since \( a \) is a rational number, \( a^3 \) will also be 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 calculated using the Pythagorean theorem: \( d = a\sqrt{2} \). Since \( \sqrt{2} \) is an irrational number, the diagonal will be irrational.
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The surface area of the cube: The surface area \( S \) of a cube is calculated using the formula \( S = 6a^2 \). Since \( a \) is a rational number, \( 6a^2 \) will be rational.
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The area of one of the cube’s faces: The area \( A \) of one face of the cube is calculated using \( A = a^2 \). Since \( a \) is a rational number, \( a^2 \) will also be 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.