Question

Стороны основания прямого параллелепипеда равны 7 см и 3корня из 2 см, а острый угол основания равен 45 градусов. Меньшая диагональ параллелепипеда составляет угол в 45 градусов с плоскостью основания. Найдите объем параллелепипеда.Геометрия 10-11 Л.С.Атанасян №728

Answer 1

Меньшая диагональ ОСНОВАНИЯ легко находится по теореме косинусов, её квадрат равен

 

7^2+(3sqrt(2))^2 - 2*7*3*sqrt(2)*sqrt(2)/2 = 25; 

 

Дальше, высота параллелепипеда, она же - боковое ребро, образует с этой меньшей диагональю ОСНОВАНИЯ длинны 5 квадрат (поскольку так получается, что в прямоугольнике угол между одной  диагональю и стороной - 45 градусов). То есть тоже равна 5. 

 

Площадь основания S = 7*3*sqrt(2)*sqrt(2)/2 = 21;

 

V = 21*5 = 105

Answer 2

Given Information:

We are given the following information about a right rectangular parallelepiped: - The sides of the base are equal to 7 cm and 3√2 cm. - The acute angle between the base and the smaller diagonal is 45 degrees.

Finding the Volume of the Parallelepiped:

To find the volume of the parallelepiped, we need to know the length of the smaller diagonal. Let's denote the length of the smaller diagonal as d.

To find the length of the smaller diagonal, we can use the Pythagorean theorem. In a right rectangular parallelepiped, the length of the smaller diagonal is the hypotenuse of a right triangle formed by the sides of the base and the height of the parallelepiped.

Let's denote the sides of the base as a = 7 cm and b = 3√2 cm. The height of the parallelepiped is the distance between the base and the opposite vertex.

Using the Pythagorean theorem, we have:

d^2 = a^2 + b^2 + h^2

Since the acute angle between the base and the smaller diagonal is 45 degrees, we can use the fact that the cosine of 45 degrees is equal to 1/√2. Therefore, we have:

h = d * cos(45) = d / √2

Substituting this into the equation above, we get:

d^2 = a^2 + b^2 + (d/√2)^2

Simplifying the equation, we have:

d^2 = a^2 + b^2 + d^2/2

Multiplying both sides of the equation by 2, we get:

2d^2 = 2a^2 + 2b^2 + d^2

Simplifying further, we have:

d^2 = 2a^2 + 2b^2

Substituting the values of a = 7 cm and b = 3√2 cm, we can solve for d:

d^2 = 2(7^2) + 2(3√2^2)

d^2 = 98 + 18

d^2 = 116

Taking the square root of both sides, we find:

d = √116

Now that we have the length of the smaller diagonal, we can find the volume of the parallelepiped. The volume of a parallelepiped is given by the formula:

Volume = length * width * height

In this case, the length is a, the width is b, and the height is h (which is equal to d/√2).

Substituting the values, we have:

Volume = a * b * (d/√2)

Volume = 7 cm * 3√2 cm * (√116/√2)

Simplifying, we have:

Volume = 21√2 cm^2 * (√116/√2)

Volume = 21√2 cm^2 * √(116/2)

Volume = 21√2 cm^2 * √58

Volume = 21√(2 * 58) cm^3

Volume = 21√116 cm^3

Volume = 21 * 2√29 cm^3

Volume = 42√29 cm^3

Therefore, the volume of the parallelepiped is 42√29 cm^3.

Answer:

The volume of the parallelepiped is 42√29 cm^3.