Question
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 surface area of the cube the surface area of the cube 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 volume of the cube
Answers
GPT-4o mini
To determine which measurement will require the use of irrational numbers, let's analyze each option:
1. **Surface Area of the Cube**: The formula for the surface area \( S \) of a cube with edge length \( a \) is given by \( S = 6a^2 \). Since \( a \) is rational, \( a^2 \) is also rational, and thus \( S \) is rational.
2. **Diagonal of One of the Cube’s Faces**: The diagonal \( d \) of a face of the cube can be computed using the Pythagorean theorem: \( d = a\sqrt{2} \). Since \( a \) is rational, \( \sqrt{2} \) is an irrational number, making \( d \) irrational.
3. **Area of One of the Cube’s Faces**: The area \( A \) of one face of the cube is given by \( A = a^2 \). Since \( a \) is rational, \( A \) is rational.
4. **Volume of the Cube**: The volume \( V \) is calculated using \( V = a^3 \). Since \( a \) is rational, \( V \) will also be rational.
Thus, the measurement that requires the use of irrational numbers is:
**The diagonal of one of the cube’s faces**.
1. **Surface Area of the Cube**: The formula for the surface area \( S \) of a cube with edge length \( a \) is given by \( S = 6a^2 \). Since \( a \) is rational, \( a^2 \) is also rational, and thus \( S \) is rational.
2. **Diagonal of One of the Cube’s Faces**: The diagonal \( d \) of a face of the cube can be computed using the Pythagorean theorem: \( d = a\sqrt{2} \). Since \( a \) is rational, \( \sqrt{2} \) is an irrational number, making \( d \) irrational.
3. **Area of One of the Cube’s Faces**: The area \( A \) of one face of the cube is given by \( A = a^2 \). Since \( a \) is rational, \( A \) is rational.
4. **Volume of the Cube**: The volume \( V \) is calculated using \( V = a^3 \). Since \( a \) is rational, \( V \) will also be rational.
Thus, the measurement that requires the use of irrational numbers is:
**The diagonal of one of the cube’s faces**.