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 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 the diagonal of one of the cube’s faces

Answer 1

To identify which measurement will require the use of irrational numbers when the cube has an edge length \( a \) that is a rational number, we can analyze each option:

1. **The area of one of the cube’s faces**: The area \( A \) of a face of the cube is given by \( A = a^2 \). Since \( a \) is rational, \( a^2 \) is also rational.

2. **The surface area of the cube**: The surface area \( S \) of the cube is \( S = 6a^2 \). Again, since \( a^2 \) is rational, \( S \) is also rational.

3. **The volume of the cube**: The volume \( V \) of the cube is \( V = a^3 \). Since \( a \) is rational, \( a^3 \) is also rational.

4. **The diagonal of one of the cube’s faces**: The diagonal \( d \) of one of the cube's faces is calculated using the Pythagorean theorem:
\[
d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2}.
\]
Since \( a \) is rational, \( d \) involves \( \sqrt{2} \), which is an irrational number.

Thus, the measurement that will require the use of irrational numbers is **the diagonal of one of the cube’s faces**.