A cube has an edge length, a
, that is a rational number of centimeters. Identify which measurement will require the use of irrational numbers.
One measurement that will require the use of irrational numbers is the diagonal of the cube.
The diagonal of a cube can be found by using the Pythagorean Theorem. Considering the cube's edge length is "a", the diagonal would be the length of the hypotenuse of a right-angled triangle with two side lengths equal to "a".
Using the Pythagorean Theorem, we have:
diagonal^2 = a^2 + a^2
diagonal^2 = 2a^2
To solve for the diagonal, we need to take the square root of both sides:
diagonal = √(2a^2)
Since the edge length "a" is given as a rational number, the square root of 2 (√2) will introduce an irrational number. Thus, the measurement of the diagonal of a cube will require the use of irrational numbers.
Which of the following numbers could be used to specify how many cups of flour should be used in a bread recipe?
Responses
square root of 3
square root of 3
−3
negative 3
2 3/4
2 and 3 fourths
47
The number that could be used to specify how many cups of flour should be used in a bread recipe is 2 3/4. This is a mixed number, representing 2 whole cups and 3/4 of a cup of flour. The other options, square root of 3, -3, and 47, are not appropriate measurements for specifying the amount of flour in cups.
How do you show the group of digits that repeat in a repeating decimal?
To show the group of digits that repeat in a repeating decimal, you typically use a horizontal line (or a vinculum) over the digits that repeat. This is called a repeating decimal bar.
For example, if you have the decimal 0.333333..., where the digit 3 repeats infinitely, you would write it as 0.3 with a repeating decimal bar over the digit 3:
0.3
Similarly, for the decimal 0.272727..., where the digits 27 repeat infinitely, you would write it as 0.2 with a repeating decimal bar over the digits 27:
0.2
The repeating decimal bar helps to show which group of digits repeats indefinitely in the decimal representation.
The measurements that will require the use of irrational numbers for a cube are:
1. Diagonal Length: The diagonal length of a cube can be found using the Pythagorean Theorem. Since the diagonal passes through the cube from one corner to the opposite corner, it creates a right triangle. The diagonal length can be calculated as the square root of the sum of the squares of two sides. Since the edge length (a) is rational, this calculation will result in an irrational number.
2. Surface Area: The surface area of a cube can be found by multiplying the area of one face by the total number of faces. Since the area of one face is the square of the edge length (a^2), this calculation does not require irrational numbers.
3. Volume: The volume of a cube can be found by cubing the edge length (a^3). Since the edge length (a) is rational, this calculation does not require irrational numbers.
Therefore, the measurement that will require the use of irrational numbers for a cube is the diagonal length.
To identify which measurement will require the use of irrational numbers in a cube with an edge length (a) that is a rational number of centimeters, we need to consider the different measurements of the cube.
1. Volume:
The volume of a cube is given by the formula V = a^3, where a is the edge length. Since a is a rational number, raising it to the power of 3 will also result in a rational number. Therefore, the volume measurement of the cube does not require irrational numbers.
2. Surface Area:
The surface area of a cube is given by the formula SA = 6a^2. Again, since a is rational, squaring it will give a rational number. Thus, the surface area measurement of the cube does not require irrational numbers either.
3. Diagonal:
The diagonal of a cube can be determined by using the Pythagorean theorem. In a cube, the diagonal connects two opposite corners (vertices) of the cube. By applying the Pythagorean theorem to the three-dimensional right triangle formed by the diagonal, the edge length, and a face diagonal, we get:
diagonal^2 = a^2 + a^2 + a^2
diagonal^2 = 3a^2
diagonal = √(3a^2)
Here, the square root of 3 in the formula for diagonal makes it an irrational number. Therefore, the diagonal measurement of the cube requires the use of an irrational number.
In conclusion, the measurement that requires the use of an irrational number in a cube with a rational edge length is the diagonal.