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The length, width, and height of a cube are all labeled a.
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 diagonal of one of the cube’s faces
the diagonal of one of the cube’s faces
the volume of the cube
the volume of the cube
the surface area of the cube
the surface area of the cube
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The length, width, and height of a cube are all labeled a.
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 diagonal of one of the cube’s faces
the diagonal of one of the cube’s faces
the volume of the cube
the volume of the cube
the surface area of the cube
the surface area of the cube
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Answers
There are no human answers yet.
Answered by
GPT-4o mini
To determine which measurement of the cube requires the use of irrational numbers, we can evaluate each option:
1. **The area of one of the cube’s faces**: The area of a square face is calculated as \( a^2 \), which, since \( a \) is a rational number, results in a rational number.
2. **The diagonal of one of the cube’s faces**: The diagonal \( d \) of a square face can be calculated using the Pythagorean theorem: \( d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \). Since \( \sqrt{2} \) is an irrational number, the diagonal will be an irrational number when \( a \) is rational.
3. **The volume of the cube**: The volume \( V \) of the cube is calculated as \( V = a^3 \), which results in a rational number if \( a \) is rational.
4. **The surface area of the cube**: The surface area \( S \) is calculated as \( S = 6a^2 \), which results in a rational number if \( a \) is 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.**
1. **The area of one of the cube’s faces**: The area of a square face is calculated as \( a^2 \), which, since \( a \) is a rational number, results in a rational number.
2. **The diagonal of one of the cube’s faces**: The diagonal \( d \) of a square face can be calculated using the Pythagorean theorem: \( d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \). Since \( \sqrt{2} \) is an irrational number, the diagonal will be an irrational number when \( a \) is rational.
3. **The volume of the cube**: The volume \( V \) of the cube is calculated as \( V = a^3 \), which results in a rational number if \( a \) is rational.
4. **The surface area of the cube**: The surface area \( S \) is calculated as \( S = 6a^2 \), which results in a rational number if \( a \) is 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.**
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