To determine which measurement will require the use of irrational numbers in a cube with a rational edge length \( a \), let's analyze each option:
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The diagonal of one of the cube’s faces: The diagonal \( d \) of one face of the cube can be calculated using the Pythagorean theorem. If the edge length of the cube is \( a \), the formula for the diagonal is: \[ d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \] Since \( a \) is rational, \( a\sqrt{2} \) is irrational (because \( \sqrt{2} \) is irrational).
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The surface area of the cube: The surface area \( SA \) of the cube is given by: \[ SA = 6a^2 \] This is rational since the product of rational numbers is rational.
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The volume of the cube: The volume \( V \) of the cube is calculated as: \[ V = a^3 \] This is also rational since the power of a rational number is rational.
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The area of one of the cube’s faces: The area \( A \) of one face is: \[ A = a^2 \] This is rational as well.
The measurement that requires the use of irrational numbers is:
The diagonal of one of the cube’s faces.
1 answer