Question
A cube has an edge length, , that is a rational number of centimeters. Identify which measurement will require
the use of irrational numbers.
(1 point)
a
the surface area of the cube
the volume of the cube
the area of one of the cube’s faces
the diagonal of one of the cube’s faces
the use of irrational numbers.
(1 point)
a
the surface area of the cube
the volume of the cube
the area of one of the cube’s faces
the diagonal of one of the cube’s faces
Answers
GPT-4o mini
To identify which measurement will require the use of irrational numbers for a cube with an edge length \( a \) that is a rational number, we will consider the following expressions:
1. **Surface Area of the Cube**: The surface area \( S \) of a cube is given by the formula \( S = 6a^2 \). Since \( a \) is rational, \( a^2 \) is also rational, and thus the surface area is rational.
2. **Volume of the Cube**: The volume \( V \) of a cube is given by the formula \( V = a^3 \). Since \( a \) is rational, \( a^3 \) is also rational, so the volume is rational.
3. **Area of One of the Cube’s Faces**: The area \( A \) of one face of the cube is given by \( A = a^2 \). As with the surface area, since \( a \) is rational, \( a^2 \) is rational, so the area of one face is rational.
4. **Diagonal of One of the Cube’s Faces**: The diagonal \( d \) of one face of the cube (which is a square) can be computed using the Pythagorean theorem: \( d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \). Since \( a \) is rational, \( d \) will be irrational because \( \sqrt{2} \) is an irrational number.
Therefore, the measurement that will require the use of irrational numbers is:
**The diagonal of one of the cube’s faces.**
1. **Surface Area of the Cube**: The surface area \( S \) of a cube is given by the formula \( S = 6a^2 \). Since \( a \) is rational, \( a^2 \) is also rational, and thus the surface area is rational.
2. **Volume of the Cube**: The volume \( V \) of a cube is given by the formula \( V = a^3 \). Since \( a \) is rational, \( a^3 \) is also rational, so the volume is rational.
3. **Area of One of the Cube’s Faces**: The area \( A \) of one face of the cube is given by \( A = a^2 \). As with the surface area, since \( a \) is rational, \( a^2 \) is rational, so the area of one face is rational.
4. **Diagonal of One of the Cube’s Faces**: The diagonal \( d \) of one face of the cube (which is a square) can be computed using the Pythagorean theorem: \( d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \). Since \( a \) is rational, \( d \) will be irrational because \( \sqrt{2} \) is an irrational number.
Therefore, the measurement that will require the use of irrational numbers is:
**The diagonal of one of the cube’s faces.**