Question
A cube has an edge length, a%0D%0A𝑎%0D%0A, that is a rational number of centimeters. Identify which measurement will require the use of irrational numbers.%0D%0A%0D%0A(1 point)%0D%0AResponses%0D%0A%0D%0Athe volume of the cube%0D%0Athe volume of the cube%0D%0A%0D%0Athe surface area of the cube%0D%0Athe surface area of the cube%0D%0A%0D%0Athe area of one of the cube’s faces%0D%0Athe area of one of the cube’s faces%0D%0A%0D%0Athe diagonal of one of the cube’s faces
Answers
GPT-4o mini
In a cube with an edge length \( a \), if \( a \) is a rational number, the volume of the cube (\( V = a^3 \)), the surface area of the cube (\( SA = 6a^2 \)), and the area of one of the cube’s faces (\( A = a^2 \)) will all be rational numbers as well, since they involve only rational operations.
However, the diagonal of one of the cube's faces can be found using the Pythagorean theorem. For a face of the cube, which is a square, the length of the diagonal \( d \) would be calculated as:
\[
d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2}
\]
Since \( \sqrt{2} \) is an irrational number, the diagonal of one of the cube’s faces involves an irrational number.
Therefore, the correct response is **the diagonal of one of the cube’s faces**.
However, the diagonal of one of the cube's faces can be found using the Pythagorean theorem. For a face of the cube, which is a square, the length of the diagonal \( d \) would be calculated as:
\[
d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2}
\]
Since \( \sqrt{2} \) is an irrational number, the diagonal of one of the cube’s faces involves an irrational number.
Therefore, the correct response is **the diagonal of one of the cube’s faces**.