When considering a cube with an edge length that is a rational number, we can analyze the different measurements provided:
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Volume of the cube: The volume \( V \) of a cube is calculated using the formula: \[ V = \text{edge length}^3 \] Since the edge length is a rational number, the volume will also be a rational number (the cube of a rational number is still rational).
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Surface area of the cube: The surface area \( SA \) of a cube is calculated using the formula: \[ SA = 6 \times \text{edge length}^2 \] Again, since the edge length is a rational number, the surface area will also be a rational number (the square of a rational number is rational, and multiplying by 6 keeps it 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: \[ A = \text{edge length}^2 \] Similar to the surface area, since the edge length is rational, the area of one face will also be a rational number.
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Diagonal of one of the cube’s faces: The diagonal \( d \) of one face of the cube can be calculated using the Pythagorean theorem: \[ d = \sqrt{\text{edge length}^2 + \text{edge length}^2} = \sqrt{2 \times \text{edge length}^2} = \text{edge length} \times \sqrt{2} \] Since \(\sqrt{2}\) is an irrational number, when you multiply it by a rational number (the edge length), the result will be irrational.
Therefore, the measurement that requires the use of irrational numbers is:
The diagonal of one of the cube’s faces.
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