To assess which measurement will require the use of irrational numbers based on a cube with an edge length \( a \) that is a rational number:
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Volume of the Cube: The volume \( V \) of a cube is given by the formula: \[ V = a^3 \] Since \( a \) is rational, \( a^3 \) will also be rational.
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Surface Area of the Cube: The surface area \( SA \) of a cube is given by the formula: \[ SA = 6a^2 \] As \( a \) 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 given by the formula: \[ A = a^2 \] Since \( a \) is rational, \( a^2 \) 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 can be calculated using the Pythagorean theorem: \[ d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \] Here, \( \sqrt{2} \) is an irrational number. Therefore, since \( a \) is rational, \( a\sqrt{2} \) is irrational.
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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