Question

Сторона куба дорівнює 10см. Обчислити діагональ бічної грані, діагональ куба і кут діагоналі куба до основи. розв'язок з малюнком . Дуже СРОЧНОО

Answer 1

Calculation of the Diagonal of a Cube's Side

To calculate the diagonal of a cube's side, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In the case of a cube, the diagonal of a side forms a right-angled triangle with two sides of equal length (the side of the cube) and the diagonal as the hypotenuse.

Let's denote the length of the side of the cube as s and the diagonal of the side as d.

According to the Pythagorean theorem, we have the following equation:

s^2 + s^2 = d^2

Since the side of the cube is given as 10 cm, we can substitute s = 10 cm into the equation:

10^2 + 10^2 = d^2

Simplifying the equation:

100 + 100 = d^2

200 = d^2

Taking the square root of both sides to solve for d:

d = √200

Calculating the square root of 200, we find that the diagonal of the cube's side is approximately 14.14 cm.

Calculation of the Diagonal of the Cube

To calculate the diagonal of the cube, we can use the Pythagorean theorem again. This time, we consider a right-angled triangle formed by the diagonal of the cube, one side of the cube, and the diagonal of a side.

Let's denote the diagonal of the cube as D.

According to the Pythagorean theorem, we have the following equation:

D^2 = d^2 + d^2

Substituting the value of d (which we calculated earlier as approximately 14.14 cm) into the equation:

D^2 = 14.14^2 + 14.14^2

Simplifying the equation:

D^2 = 200 + 200

D^2 = 400

Taking the square root of both sides to solve for D:

D = √400

Calculating the square root of 400, we find that the diagonal of the cube is 20 cm.

Calculation of the Angle between the Diagonal of the Cube and the Base

To calculate the angle between the diagonal of the cube and the base, we can use trigonometry. Specifically, we can use the inverse tangent function (arctan) to find the angle.

Let's denote the angle between the diagonal of the cube and the base as θ.

Using the right-angled triangle formed by the diagonal of the cube, one side of the cube, and the diagonal of a side, we can set up the following equation:

tan(θ) = (d / s)

Substituting the values of d (approximately 14.14 cm) and s (10 cm) into the equation:

tan(θ) = (14.14 / 10)

Calculating the arctan of both sides to solve for θ:

θ = arctan(14.14 / 10)

Using a calculator, we find that the angle between the diagonal of the cube and the base is approximately 54.74 degrees.

I hope this helps! Let me know if you have any further questions.