Question

боковое ребро правильной треугольной пирамиды образует с плоскостью основания угол arctg0,5 радиус окружности вписанной в основание пирамиды равен 6 см правильно основанию проведено сечение пирамиды площадь которого в 9 раз меньше площади основания в результате получилось 2 многоугольника. Найдите объём большего из них.

Answer 1

У правильной треугольной пирамиды есть свойство: тангенс угла αнаклона боковой грани к основанию в 2 раза больше тангенса угла β наклона бокового ребра.

Тогда tgα = 2*0,5 = 1, поэтому угол наклона α = 45 градусов.

Из этого следует, что высота Н пирамиды равна радиусу вписанной окружности: Н = 6 см.

По радиусу r вписанной окружности находим длину ребра а основания как равностороннего треугольника.

a = 2*r/tg 30° = 2*6/(1/√3) = 12√3 см.

Площадь основания So = a²√3/4 = 144*3√3/4 = 108√3 см².

Находим объём пирамиды: V = (1/3)SoH = (1/3)*(108√3)*6 = 216√3 см³.

По условию параллельно основанию проведено сечение пирамиды, площадь которого в 9 раз меньше площади основания.

По свойству подобия фигур радиус вписанной окружности в основание сечения в корень из 9 раз меньше заданного радиуса 6.

А объём пропорционален кубу коэффициента подобия.

Значит, объём V1 отсечённой части пирамид равен:

V1 = (1/3³)V = (1/27)*(216√3) = 8√3 см³.

Ответ: объём большего многоугольника равен 216√3 - 8√3 = 208√3 см³.

Answer 2

Problem Analysis

To find the volume of the larger polygon resulting from the section of the pyramid, we need to determine the side length of the base of the pyramid and the height of the pyramid. With this information, we can calculate the volume of the pyramid and then subtract the volume of the smaller polygon to find the volume of the larger polygon.

Given Information

- The angle between the lateral edge of the pyramid and the base plane is arctan(0.5) radians. - The radius of the inscribed circle in the base of the pyramid is 6 cm. - The area of the section of the pyramid is 9 times smaller than the area of the base.

Solution Steps

1. Find the side length of the base of the pyramid. 2. Find the height of the pyramid. 3. Calculate the volume of the pyramid. 4. Calculate the area of the smaller polygon. 5. Calculate the volume of the smaller polygon. 6. Subtract the volume of the smaller polygon from the volume of the pyramid to find the volume of the larger polygon.

Let's calculate each step in detail.

Step 1: Finding the Side Length of the Base

To find the side length of the base of the pyramid, we can use the radius of the inscribed circle. The radius of the inscribed circle is equal to half the length of the side of the base of the pyramid. Therefore, the side length of the base can be calculated as follows:

Side length of the base = 2 * radius of the inscribed circle

Substituting the given value of the radius (6 cm), we get:

Side length of the base = 2 * 6 cm = 12 cm [[1]]

Step 2: Finding the Height of the Pyramid

To find the height of the pyramid, we can use the trigonometric relationship between the height, the side length of the base, and the angle between the lateral edge and the base plane. The relationship is given by:

Height = side length of the base * tan(angle)

Substituting the given value of the side length of the base (12 cm) and the angle (arctan(0.5) radians), we get:

Height = 12 cm * tan(arctan(0.5))

Using the identity tan(arctan(x)) = x, we simplify the equation to:

Height = 12 cm * 0.5 = 6 cm [[2]]

Step 3: Calculating the Volume of the Pyramid

The volume of a pyramid can be calculated using the formula:

Volume = (1/3) * base area * height

The base area of a regular triangular pyramid is given by:

Base area = (sqrt(3)/4) * (side length of the base)^2

Substituting the given values of the side length of the base (12 cm) and the height (6 cm), we get:

Base area = (sqrt(3)/4) * (12 cm)^2

Simplifying the equation, we find:

Base area = 36 * sqrt(3) cm^2

Now, substituting the calculated values of the base area and the height into the volume formula, we get:

Volume = (1/3) * (36 * sqrt(3) cm^2) * 6 cm

Simplifying the equation, we find:

Volume = 72 * sqrt(3) cm^3 [[3]]

Step 4: Calculating the Area of the Smaller Polygon

The area of the smaller polygon resulting from the section of the pyramid is given as 9 times smaller than the area of the base. Therefore, we can calculate the area of the base and then divide it by 9 to find the area of the smaller polygon.

Area of the smaller polygon = (Area of the base) / 9

Substituting the calculated value of the base area (36 * sqrt(3) cm^2), we get:

Area of the smaller polygon = (36 * sqrt(3) cm^2) / 9

Simplifying the equation, we find:

Area of the smaller polygon = 4 * sqrt(3) cm^2 [[4]]

Step 5: Calculating the Volume of the Smaller Polygon

To calculate the volume of the smaller polygon, we need to multiply the area of the smaller polygon by the height of the pyramid. Using the calculated value of the height (6 cm), we get:

Volume of the smaller polygon = (Area of the smaller polygon) * height

Substituting the calculated value of the area of the smaller polygon (4 * sqrt(3) cm^2) and the height (6 cm), we get:

Volume of the smaller polygon = (4 * sqrt(3) cm^2) * 6 cm

Simplifying the equation, we find:

Volume of the smaller polygon = 24 * sqrt(3) cm^3 [[5]]

Step 6: Finding the Volume of the Larger Polygon

To find the volume of the larger polygon, we subtract the volume of the smaller polygon from the volume of the pyramid.

Volume of the larger polygon = Volume of the pyramid - Volume of the smaller polygon

Substituting the calculated values of the volume of the pyramid (72 * sqrt(3) cm^3) and the volume of the smaller polygon (24 * sqrt(3) cm^3), we get:

Volume of the larger polygon = (72 * sqrt(3) cm^3) - (24 * sqrt(3) cm^3)

Simplifying the equation, we find:

Volume of the larger polygon = 48 * sqrt(3) cm^3

Therefore, the volume of the larger polygon is 48 * sqrt(3) cm^3.