Use the image to answer the 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 diagonal of one of the cube’s faces
the diagonal of one of the cube’s faces

the volume of the cube

5 answers

To determine which measurement will require the use of irrational numbers for a cube with edge length \( a \) (a rational number), let's analyze each option:

  1. The area of one of the cube’s faces: The area 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 a cube is calculated as \( S = 6a^2 \). Again, since \( a \) is rational, \( a^2 \) is rational, and thus \( 6a^2 \) is also rational.

  3. The diagonal of one of the cube’s faces: The diagonal \( d \) of one of the cube's faces can be found using the Pythagorean theorem: \( d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \). Since \( \sqrt{2} \) is an irrational number, \( a\sqrt{2} \) is irrational (as it is the product of a rational number \( a \) and an irrational number \( \sqrt{2} \)).

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

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

The diagonal of one of the cube’s faces.

Pick an answer I said

The measurement that will require the use of irrational numbers is:

the diagonal of one of the cube’s faces.

Now bot is this right🤨

Yes, that is correct. The diagonal of one of the cube's faces, given by the formula \( d = a\sqrt{2} \), requires the use of an irrational number because \( \sqrt{2} \) is irrational. Thus, \( a\sqrt{2} \) becomes irrational if \( a \) is a rational number.

If you have any other questions or need further clarification, feel free to ask!