Question

Боковое ребро правильной четырехугольной призмы 6 см,а диагональ боковой грани 10 см.найдите боковую поверхность и объем призмы!ребят решение подробной,пжлст=)

Answer 1

найдем сторону основания: на боковой грани сторона основания, высота и диагональ образуют прямоугольный треугольник, применим теорему Пифагора -

сторона основания = корень (100-36)=8

Тогда Sбок=Pосн*h=32*6=192;   Vпризмы=Sосн*h=64*6=384 

Answer 2

Problem Analysis

We are given the length of one of the lateral edges of a regular quadrilateral prism, which is 6 cm, and the diagonal of one of the lateral faces, which is 10 cm. We need to find the lateral surface area and the volume of the prism.

Finding the Lateral Surface Area

To find the lateral surface area of the prism, we need to find the perimeter of the base and multiply it by the height of the prism.

The base of the prism is a regular quadrilateral, which means all its sides are equal in length. Since we know the length of one of the lateral edges is 6 cm, we can conclude that all the sides of the base are also 6 cm long.

To find the perimeter of the base, we multiply the length of one side by the number of sides, which is 4 in this case. Therefore, the perimeter of the base is 6 cm * 4 = 24 cm.

Now, we need to find the height of the prism. We are given the diagonal of one of the lateral faces, which is 10 cm. Since the diagonal of a quadrilateral divides it into two congruent right triangles, we can use the Pythagorean theorem to find the height.

Let's assume the height of the prism is h cm. Using the Pythagorean theorem, we have:

h^2 = (6 cm)^2 - (5 cm)^2

Simplifying the equation, we get:

h^2 = 36 cm^2 - 25 cm^2

h^2 = 11 cm^2

Taking the square root of both sides, we find:

h = √11 cm

Now that we have the height, we can calculate the lateral surface area by multiplying the perimeter of the base by the height:

Lateral Surface Area = Perimeter of Base * Height = 24 cm * √11 cm

Therefore, the lateral surface area of the prism is 24√11 cm^2.

Finding the Volume

To find the volume of the prism, we multiply the area of the base by the height.

The base of the prism is a regular quadrilateral, which means it is a square. The area of a square is given by the formula:

Area of Base = (Side Length)^2

Since we know the side length of the base is 6 cm, we can calculate the area of the base as:

Area of Base = (6 cm)^2 = 36 cm^2

Now, we multiply the area of the base by the height to find the volume:

Volume = Area of Base * Height = 36 cm^2 * √11 cm

Therefore, the volume of the prism is 36√11 cm^3.

Summary

- The lateral surface area of the prism is 24√11 cm^2. - The volume of the prism is 36√11 cm^3.

Please note that the calculations provided are based on the given information and assumptions made.