Question

Стороны основания прямого параллелепипеда равны 3 м и 9 м, а его диагонали составляют с плоскостью основания углы в 45° и 60°. Найдите диагонали параллелепипеда, его боковую поверхность и объем

Answer 1

Ответ
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Answer 2

Given Information:

We are given that the sides of a rectangular parallelepiped are 3 m and 9 m, and its diagonals form angles of 45° and 60° with the plane of the base.

Finding the Diagonals:

To find the diagonals of the parallelepiped, we can use the given information about the angles formed by the diagonals with the plane of the base.

Let's denote the length of the diagonal that forms a 45° angle with the base as d1 and the length of the diagonal that forms a 60° angle with the base as d2.

Using trigonometry, we can relate the lengths of the diagonals to the sides of the parallelepiped and the angles formed with the base.

From the given information, we have the following relationships: - The diagonal forming a 45° angle with the base is equal to the square root of the sum of the squares of the sides: d1 = √(3^2 + 9^2) = √90. - The diagonal forming a 60° angle with the base is equal to the side length multiplied by the square root of 3: d2 = 9√3.

Therefore, the diagonals of the parallelepiped are √90 and 9√3.

Finding the Surface Area:

To find the surface area of the parallelepiped, we need to calculate the areas of its six faces.

The parallelepiped has three pairs of opposite faces, each with the same area. Let's denote the length, width, and height of the parallelepiped as l, w, and h, respectively.

The surface area of the parallelepiped can be calculated using the formula: Surface Area = 2lw + 2lh + 2wh.

From the given information, we have l = 3 m, w = 9 m, and h = ? (unknown).

To find the height, we can use the Pythagorean theorem, which states that the square of the length of the diagonal is equal to the sum of the squares of the other two sides.

Using the known sides of the parallelepiped, we can write the following equation: d1^2 = l^2 + w^2 + h^2.

Substituting the values, we have: (√90)^2 = 3^2 + 9^2 + h^2.

Simplifying the equation, we get: 90 = 9 + 81 + h^2.

Solving for h^2, we have: h^2 = 90 - 9 - 81 = 0.

Taking the square root of both sides, we find that h = 0.

Since the height is zero, the parallelepiped degenerates into a rectangle. In this case, the surface area is simply the sum of the areas of the six faces, which is equal to: Surface Area = 2lw + 2lh + 2wh = 2(3)(9) + 2(3)(0) + 2(9)(0) = 54 + 0 + 0 = 54 square meters.

Therefore, the surface area of the parallelepiped is 54 square meters.

Finding the Volume:

To find the volume of the parallelepiped, we can use the formula: Volume = length × width × height.

From the given information, we have l = 3 m, w = 9 m, and h = 0 m.

Substituting the values, we have: Volume = 3 × 9 × 0 = 0 cubic meters.

Therefore, the volume of the parallelepiped is 0 cubic meters.

Summary:

- The diagonals of the parallelepiped are √90 and 9√3. - The surface area of the parallelepiped is 54 square meters. - The volume of the parallelepiped is 0 cubic meters.

Please let me know if there's anything else I can help you with!