Question

A cube has an edge length, a. Length, width, and height
, 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 surface area of the cube

the volume of the cube

the area of one of the cube’s faces

Answer 1

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:

1. **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).

2. **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.

3. **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.

4. **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.**