Question

В прямом параллелепипеде стороны основания длиной 3 см и 4 см составляют угол в 60 градусов, а боковое ребро есть среднее пропорциональное между сторонами основания. Определить диагонали этого параллелепипеда.

Answer 1

Решение в скане........

Answer 2

Given Information:

We are given the following information about a parallelepiped: - The lengths of the sides of the base are 3 cm and 4 cm. - The angle between the sides of the base is 60 degrees. - The length of the lateral edge is the mean proportional between the sides of the base.

Solution:

To determine the diagonals of the parallelepiped, we need to find the lengths of the diagonals of the base and the length of the diagonal connecting opposite vertices of the parallelepiped.

Let's start by finding the lengths of the diagonals of the base.

Using the given information, we can determine the lengths of the diagonals of the base using the Pythagorean theorem.

Let's denote the sides of the base as a = 3 cm and b = 4 cm.

The length of the diagonal of the base can be found using the formula:

diagonal of the base = sqrt(a^2 + b^2)

Substituting the values, we get:

diagonal of the base = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5 cm.

Now, let's find the length of the diagonal connecting opposite vertices of the parallelepiped.

Since the angle between the sides of the base is 60 degrees, we can use trigonometry to find the length of the diagonal.

Let's denote the length of the lateral edge as c.

Using the given information, we know that c is the mean proportional between a and b.

The mean proportional between a and b can be found using the formula:

mean proportional = sqrt(a * b)

Substituting the values, we get:

mean proportional = sqrt(3 * 4) = sqrt(12).

Therefore, c = sqrt(12) cm.

To find the length of the diagonal connecting opposite vertices, we can use the formula:

diagonal = sqrt(a^2 + b^2 + c^2)

Substituting the values, we get:

diagonal = sqrt(3^2 + 4^2 + (sqrt(12))^2) = sqrt(9 + 16 + 12) = sqrt(37) cm.

Therefore, the length of the diagonal connecting opposite vertices of the parallelepiped is sqrt(37) cm.

To summarize: - The length of the diagonal of the base is 5 cm. - The length of the diagonal connecting opposite vertices of the parallelepiped is sqrt(37) cm.

Please let me know if I can help you with anything else.