The Real Number System Quick Check
4 of 54 of 5 Items
Question
Use the image to answer the question.
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 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
the area of one of the cube’s faces
the area of one of the cube’s faces
Which list shows an accurate hierarchy of number sets within the real number system, from largest to smallest?(1 point)
Responses
natural numbers, whole numbers, integers, rational numbers
natural numbers, whole numbers, integers, rational numbers
integers, rational numbers, natural numbers, whole numbers
integers, rational numbers, natural numbers, whole numbers
irrational numbers, square roots, endless decimals, pi
irrational numbers, square roots, endless decimals, pi
rational numbers, integers, whole numbers, natural numbers
rational numbers, integers, whole numbers, natural numbers
rational numbers, integers, whole numbers, natural numbers
The measurement that will require the use of irrational numbers is the diagonal of one of the cube's faces.
To determine which measurement will require the use of irrational numbers, we need to understand the properties of a cube.
A cube is a three-dimensional shape with six equal square faces. Each face of the cube has side length a, which is a rational number.
Now let's analyze each option to see which one requires the use of irrational numbers:
1. The diagonal of one of the cube's faces: A diagonal is the line segment connecting two nonadjacent corners of a square. In this case, the diagonal will pass through the face of the cube. To find the diagonal of a square face, we can use the Pythagorean theorem, which involves taking the square root of the sum of the squares of the side lengths. Since the side length, a, is a rational number, the diagonal will also result in a rational number.
2. The volume of the cube: The volume of a cube is calculated by multiplying the length, width, and height of the cube. Since all three dimensions are labeled as a, and a is a rational number, the volume of the cube will also be a rational number.
3. The surface area of the cube: The surface area of a cube is found by multiplying the area of one face by six, as there are six equal faces. The area of one face is calculated by squaring the side length, a^2. Since a is a rational number, the surface area of the cube will also be a rational number.
4. The area of one of the cube's faces: As mentioned before, the area of one face of a cube is calculated by squaring the side length, a^2. Since a is given as a rational number, the area of the face will also be a rational number.
After analyzing all the options, the only measurement that will require the use of irrational numbers is the diagonal of one of the cube's faces. Thus, the correct answer is "the diagonal of one of the cube's faces."