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Question

The length, width, and height of a cube are all labeled a.

A cube has an edge length, a , that is a rational number of centimeters. Identify which measurement will require the use of irrational numbers.

(1 point)
Responses

the diagonal of one of the cube’s faces
the diagonal of one of the cube’s faces

the area of one of the cube’s faces
the area of one of the cube’s faces

the surface area of the cube
the surface area of the cube

the volume of the cube
the volume of the cube
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Answer 1

To determine which measurement will require the use of irrational numbers, we need to analyze each option:

1. The diagonal of one of the cube’s faces - The formula for the diagonal \(d\) of a square face (with side length \(a\)) is given by \(d = a\sqrt{2}\). Since \(a\) is a rational number, \(d\) will be an irrational number.

2. The area of one of the cube’s faces - The area \(A\) of a square face is given by \(A = a^2\). Since \(a\) is rational, \(A\) will also be rational.

3. The surface area of the cube - The surface area \(SA\) of a cube is given by \(SA = 6a^2\). Since \(a^2\) is rational (as \(a\) is rational), the surface area will be rational.

4. The volume of the cube - The volume \(V\) of a cube is given by \(V = a^3\). Since \(a\) is rational, \(V\) will also be rational.

Based on this analysis, the measurement that requires the use of irrational numbers is:

The diagonal of one of the cube’s faces.