To determine which measurement will require the use of irrational numbers, we can analyze each option given the edge length \( a \):
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The diagonal of one of the cube’s faces: The diagonal of a face of a cube can be calculated using the Pythagorean theorem. For a face with side length \( a \), the diagonal (d) is given by: \[ d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \] Since \( a \) is a rational number, \( a\sqrt{2} \) is irrational because \( \sqrt{2} \) is irrational.
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The surface area of the cube: The surface area (SA) of a cube is calculated as: \[ SA = 6a^2 \] This is a rational number, as the square of a rational number ( \( a^2 \) ) is rational, and multiplying by 6 keeps it rational.
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The volume of the cube: The volume (V) of a cube is calculated as: \[ V = a^3 \] This is also a rational number because the cube of a rational number is rational.
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The area of one of the cube’s face: The area (A) of one face is: \[ A = a^2 \] Again, this is a rational number since the square of a rational number is rational.
Based on this analysis, the only measurement that requires the use of irrational numbers is:
The diagonal of one of the cube’s faces.
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