A cube has an edge length ,a, that is a rational number of centimeters. Identify which measurement will require the use of irrational numbers.
A. the surface area of the cube
B. the diagonal of one of the cube’s faces
C. the volume of the cube
D. the area of one of the cube’s faces
The measurement that will require the use of irrational numbers is:
B. the diagonal of one of the cube's faces.
The measurement that will require the use of irrational numbers is:
B. the diagonal of one of the cube's faces.
To calculate the diagonal of one of the cube's faces, we can use the Pythagorean theorem. Let's assume the edge length of the cube is "a."
Using the Pythagorean theorem:
Diagonal^2 = (Edge)^2 + (Edge)^2
Diagonal^2 = a^2 + a^2
Diagonal^2 = 2a^2
To find the diagonal, we need to take the square root of both sides:
Diagonal = √(2a^2)
Since the edge length "a" is rational, the value inside the square root is 2 times a rational number. Therefore, the diagonal of one of the cube's faces will require the use of an irrational number.
To identify which measurement will require the use of irrational numbers in the given scenario, let's break down the options:
A. The surface area of the cube:
To calculate the surface area of a cube, you need to find the square of the edge length and multiply it by 6. Since the edge length is given as a rational number, squaring it will result in a rational number as well. Multiplying a rational number by 6 will still yield a rational number. Hence, the surface area of the cube can be calculated using rational numbers.
B. The diagonal of one of the cube's faces:
To find the diagonal of one of the cube's faces, we can use the Pythagorean theorem. By considering a right triangle with the edge of the cube as one side and the diagonal of the face as the hypotenuse, we have two sides of equal length (each equal to the edge length). From the Pythagorean theorem, we know that the square of the hypotenuse (diagonal) is equal to the sum of the squares of the two legs (edges).
Let the edge length of the cube be 'a'. So, the diagonal, 'd', can be found as:
d^2 = a^2 + a^2
d^2 = 2a^2
d = sqrt(2a^2)
d = a * sqrt(2)
Here, we see that the diagonal of one of the cube's faces involves multiplying the edge length by the square root of 2. Since the square root of 2 is an irrational number, the diagonal length of the cube's face will also be irrational.
C. The volume of the cube:
The volume of a cube can be calculated by cubing the edge length. Since the edge length is given as a rational number, cubing it will result in a rational number as well. Hence, the volume of the cube can also be calculated using rational numbers.
D. The area of one of the cube's faces:
The area of one of the cube's faces can be found by squaring the length of one of its edges. Since the edge length is given as a rational number, squaring it will result in a rational number as well. Hence, the area of one of the cube's faces can be calculated using rational numbers.
Therefore, the measurement that requires the use of irrational numbers is option B, which is the diagonal of one of the cube's faces.