To determine which measurement will require the use of irrational numbers, let's examine each option:
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Surface area of the cube: The surface area \( S \) of a cube with edge length \( a \) is calculated as \( S = 6a^2 \). Since \( a \) is rational, \( a^2 \) is also rational, and thus the surface area is rational.
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Area of one of the cube’s faces: The area \( A \) of one face of the cube is calculated as \( A = a^2 \). Again, since \( a \) is rational, \( a^2 \) is rational, so the area of one face is also rational.
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Diagonal of one of the cube’s faces: The diagonal \( d \) of one face of the cube (which is a square) can be calculated using the Pythagorean theorem: \( d = a\sqrt{2} \). Since \( \sqrt{2} \) is irrational and \( a \) is rational, the product \( a\sqrt{2} \) is irrational. Therefore, the diagonal of one of the cube's faces requires the use of irrational numbers.
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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, so the volume is rational.
Based on the analysis, the measurement that requires the use of irrational numbers is:
the diagonal of one of the cube’s faces.
1 answer