Question

Основанием прямоугольного параллелепипеда служит квадрат со стороной 4 см. Диагональ параллелепипеда равно 4√6 см.Найти высоту параллелепипеда и угол межу диагональю и плоскостью основания.)

Answer 1

В прямоугольном параллелепипеде квадрат его диагонали равен сумме квадратов трех измерений. высота парал. будет третьим неизвестным измерением. два других - 4 и4.
(4V6)^2=4^2+4^2+h^2
h^2=16*6-32=96-32=64
h=8
высота =8
угол найдем из прямоугольного треуг., где высота -противолежащий катет, а диагональ паралл.- гипотенуза
sinα=h/c=8/4V6=2/V6=2V6/6=V6/3≈0.8163
α=54 градуса 43 минуты 7 секунд

Answer 2

Given Information

We are given that the base of the rectangular parallelepiped is a square with a side length of 4 cm. The diagonal of the parallelepiped is equal to 4√6 cm. We need to find the height of the parallelepiped and the angle between the diagonal and the plane of the base.

Finding the Height of the Parallelepiped

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

Let's denote the height of the parallelepiped as h. The side length of the base is given as 4 cm, and the diagonal of the base is also given as 4√6 cm.

Using the Pythagorean theorem, we have:

h^2 + 4^2 = (4√6)^2

Simplifying the equation:

h^2 + 16 = 96

h^2 = 96 - 16

h^2 = 80

Taking the square root of both sides:

h = √80

Simplifying further:

h = 4√5

Therefore, the height of the parallelepiped is 4√5 cm.

Finding the Angle between the Diagonal and the Plane of the Base

To find the angle between the diagonal and the plane of the base, we can use trigonometry. The angle can be determined by finding the cosine of the angle between the diagonal and the height of the parallelepiped.

Let's denote the angle between the diagonal and the height as θ.

Using the cosine function, we have:

cos(θ) = h / (4√6)

Substituting the value of h we found earlier:

cos(θ) = (4√5) / (4√6)

Simplifying the equation:

cos(θ) = √(5/6)

To find the value of θ, we can take the inverse cosine (arccos) of both sides:

θ = arccos(√(5/6))

Using a calculator, we can find the approximate value of θ:

θ ≈ 0.7227 radians

Therefore, the angle between the diagonal and the plane of the base is approximately 0.7227 radians.

Summary

- The height of the parallelepiped is 4√5 cm. - The angle between the diagonal and the plane of the base is approximately 0.7227 radians.