Question

Основанием прямого параллелепипеда служит параллелограмм со сторонами 3 см. и 5 см. Острый угол параллелограмма равен 60 градусов. Площадь большого диагонального сечения равна 63 см квадратным. Найдите площадь полной поверхности параллелепипеда.

Answer 1

начнем с основания:

дана площадь большего диагонального сечения равная 63 см.

найдем большую диагональ основания по теореме косинусов:

d1² = 3²+ 5² - 2* 3* 5 * cos(120) = 9 + 25 + 15 =49

d1 = 7

Sдиаг.сеч = d1 * h

7h = 63, h = 9

найдем площадь основания по формуле:

Sосн = ab*sina , где а и b стороны параллелограмма, sina угол между ними

Sосн = 3 * 5 * √3/2 = 15√3/2

теперь найдем S одной боковой грани, так как фигура прямая , то противоположные грани будут равны:

S1бок = 3 * 9 = 27

S2бок = 5*9 = 45

Sполн = 2Sосн + Sбок

2Sосн = 15√3

Sбок = 2S1бок + 2S2бок = 2*27 + 2*45 = 144 см²

S полн = 144 + 15√3

 

Answer 2

Given Information:

We are given that the base of the parallelepiped is a parallelogram with sides measuring 3 cm and 5 cm, and that the acute angle of the parallelogram is 60 degrees. The area of the large diagonal section is 63 square cm.

Finding the Height of the Parallelepiped:

To find the height of the parallelepiped, we can use the formula for the area of a parallelogram: Area = base * height. In this case, the base is the side length of the parallelogram, which is 3 cm, and the area is given as 63 square cm. Therefore, we can solve for the height as follows:

63 = 3 * height

Simplifying the equation, we find:

height = 63 / 3 = 21 cm

Finding the Area of the Full Surface of the Parallelepiped:

The full surface area of a parallelepiped can be calculated using the formula: 2 * (base area + side area1 + side area2).

The base area is the area of the parallelogram, which can be calculated using the formula: base area = base * height. In this case, the base area is:

base area = 3 cm * 21 cm = 63 cm^2

The side areas can be calculated using the formula: side area = base * side length. In this case, the side lengths are 5 cm and 3 cm. Therefore, the side areas are:

side area1 = 3 cm * 5 cm = 15 cm^2

side area2 = 3 cm * 5 cm = 15 cm^2

Now we can calculate the full surface area of the parallelepiped:

full surface area = 2 * (base area + side area1 + side area2)

full surface area = 2 * (63 cm^2 + 15 cm^2 + 15 cm^2)

full surface area = 2 * (63 cm^2 + 30 cm^2)

full surface area = 2 * 93 cm^2

full surface area = 186 cm^2

Therefore, the area of the full surface of the parallelepiped is 186 square cm.

Summary:

The height of the parallelepiped is 21 cm, and the area of the full surface of the parallelepiped is 186 square cm.