Question

Determine the angle between the diagonal of a cube and the diagonal of its base, as shown in the figure, where a = 12. Round your answer to one decimal place.

Answer 1

The angle between the diagonal of a cube and the diagonal of its base is 60 degrees.

Step-by-step explanation:

The angle between the diagonal of a cube and the diagonal of its base can be determined using the Pythagorean theorem. Let the length of the side of the cube be represented by 'a'. The diagonal of the base is the side length of the cube, which is 'a'. The diagonal of the cube can be found using the formula d = sqrt(3a^2). Therefore, the angle between the diagonals is given by the inverse tangent of the ratio of the diagonals, which is arctan(sqrt(3)) = 60 degrees.

Answer 2

The angle between the cube's body diagonal and the base diagonal can be found using trigonometry by applying the cosine function to the right triangle formed by the cube's side, base diagonal, and body diagonal.

Step-by-step explanation:

Calculating the angle between the diagonal of a cube and the diagonal of its base involves understanding the geometry of a cube and the relationships between its edges and diagonals. Given that the side of the cube (a) is 12 units long, we begin by finding the length of the base diagonal using the Pythagorean theorem. Since the base of the cube is a square, the diagonal's length is a√2, or approximately 16.97 units.

The cube's body diagonal forms a right triangle with the cube's side and the base's diagonal, with the body diagonal serving as the hypotenuse. We use the Pythagorean theorem again to find the length of the body diagonal, which is a√3, or approximately 20.78 units.Using these lengths and the knowledge that the triangle formed is right-angled, we can apply trigonometric functions to find the required angle.

Specifically, we use the cosine function, which relates the adjacent side (base diagonal) to the hypotenuse (body diagonal) in a right triangle. Cosine of the angle (θ) is equal to adjacent/hypotenuse, resulting in θ = cos-1(a√2 / a√3). Plugging in the value of a to this equation gives us the angle, which upon calculating and rounding to one decimal place can provide us with our final answer.