A cube has a edge length, a, that is a rational number of centimeters. identify which measurement will require the use of irrational numbers.
The measurements that will require the use of irrational numbers are the diagonal of the cube and the surface area of the cube.
1. Diagonal of the cube:
To calculate the diagonal of a cube, we need to use the Pythagorean theorem. Each of the three sides of the cube is of length 'a'. So, the length of the diagonal (d) can be found using the equation d² = a² + a² + a² = 3a². This means that the diagonal, d, of the cube will have the square root of 3 in it, which is an irrational number (√3 ≈ 1.732).
2. Surface Area of the cube:
The surface area of a cube is found by calculating the area of each of its six faces. Since each face is a square, we can find the area of one face by squaring the length of the edge (a²). Thus, the total surface area of the cube will be 6a². Since the edge length (a) is given as a rational number, we do not require irrational numbers to calculate the surface area.
Hence, the measurement of the diagonal of the cube is the one that requires the use of an irrational number (√3).