Question

Дан прямой параллелепипед, у которого стороны основания, равные 16 и 12 см, составляют угол 60 град., а боковое ребро есть средняя пропорциональная между сторонами основания. Найдите диагонали параллелепипеда.

Answer 1

Рассмотрим сначала основание, в основании лежит параллелограмм, найдем диагонали p1 и p2 этого параллелограмма. По теореме косинусов:
$p_1^2 = 16^2 + 12^2 - 2 \cdot 16 \cdot 12 \cdot \cos(60^{\circ}) =$
$= 256 + 144 - 2 \cdot 16 \cdot 12 \cdot \frac{1}{2} =$
$= 400 - 16\cdot 12$
$p_2^2 = 16^2 + 12^2 - 2\cdot 16 \cdot 12 \cdot \cos(120^{\circ}) =$
$= 256 + 144 - 2 \cdot 16 \cdot 12 \cdot (-\frac{1}{2}) =$
$= 256 + 144 + 16 \cdot 12 = 400 + 16 \cdot 12$.
Т.к. дан прямой параллелепипед, то боковые ребра перпендикулярны основаниям этого параллелепипеда. Диагонали параллелепипеда найдем по теореме Пифагора. По условию, высота параллелепипеда
$h^2 = 16 \cdot 12$ (высота есть среднее пропорциональное между сторонами основания).
$d_1^2 = p_1^2 + h^2 = 400 - 16 \cdot 12 + 16 \cdot 12 = 400$
$d_1 = \sqrt{400} = 20$
$d_2^2 = p_2^2 + h^2 = 400 + 16 \cdot 12 + 16 \cdot 12 =$
$= 400+ 2 \cdot 16 \cdot 12 = 400 + 384 = 784$
$d_2 = \sqrt{784} = 28$.
Ответ. 20 и 28.

Answer 2

Given Information

We are given a rectangular parallelepiped with base sides measuring 16 cm and 12 cm, which form a 60-degree angle. We are also told that one of the lateral edges is the mean proportional between the base sides.

Solution

To find the diagonals of the parallelepiped, we need to determine the lengths of the three edges that meet at a single vertex. Let's denote these edges as a, b, and c.

From the given information, we know that the base sides of the parallelepiped form a 60-degree angle. This means that the ratio of the longer side to the shorter side is equal to the square root of 3. Therefore, we can write:

a / b = √3 We are also told that one of the lateral edges is the mean proportional between the base sides. Let's assume that this lateral edge is b. This means that:

b / c = c / a We can use these two equations to solve for a, b, and c.

Solving for a, b, and c

From equation we have:

a = √3b Substituting this into equation we get:

(√3b) / c = c / (√3b)

Cross-multiplying, we have:

c^2 = (√3b)^2

Simplifying, we get:

c^2 = 3b^2

Taking the square root of both sides, we have:

c = √(3b^2)

Simplifying further, we get:

c = √3 * b Now, let's solve for b using equation:

a / b = √3

Rearranging, we have:

b = a / √3

Substituting this into equation we get:

c = √3 * (a / √3)

Simplifying, we have:

c = a From equation we can see that c = a. Therefore, the three edges meeting at a single vertex are equal in length.

Diagonals of the Parallelepiped

To find the diagonals of the parallelepiped, we need to find the lengths of the three edges meeting at a single vertex. Since these edges are equal in length, we can denote their length as x.

From the given information, we know that the base sides of the parallelepiped measure 16 cm and 12 cm. Therefore, we have:

a = 16 cm b = 12 cm c = a = 16 cm

Since a = c = 16 cm, we can denote the length of the three edges meeting at a single vertex as x = 16 cm.

Now, to find the diagonals of the parallelepiped, we can use the Pythagorean theorem. The diagonal of the base face is the hypotenuse of a right triangle with sides measuring a and b. Therefore, the length of the diagonal of the base face is:

d_base = √(a^2 + b^2)

Substituting the values of a and b, we have:

d_base = √(16^2 + 12^2) = √(256 + 144) = √400 = 20 cm

The diagonal of the base face is 20 cm.

The diagonal of the parallelepiped is the hypotenuse of a right triangle with sides measuring d_base and c. Therefore, the length of the diagonal of the parallelepiped is:

d_diagonal = √(d_base^2 + c^2)

Substituting the values of d_base and c, we have:

d_diagonal = √(20^2 + 16^2) = √(400 + 256) = √656 ≈ 25.61 cm

Therefore, the diagonals of the parallelepiped are approximately 20 cm and 25.61 cm.

Answer

The diagonals of the parallelepiped are approximately 20 cm and 25.61 cm.