Question

СРОЧНО ДАЮ 100б через вершину конуса проведено площину,яка відтинає від кола його основи його чверть.Радіус основи конуса дорівнює 3см а кут при вершині отриманого перерізу 60°.Знайти площу утвореного перерізу ,площу повної поверхні конуса та його об’єм

Answer 1

Ответ:

60÷3=20

60÷20=3

20-3=18

Answer 2

Ответ:

правильный ответ 16

Объяснение:

Answer 3

Problem Analysis

We are given a cone with a radius of 3 cm and a cutting plane passing through its vertex. The cutting plane divides the cone's base into a quarter of a circle. The angle at the vertex of the resulting cross-section is 60°. We need to find the area of the cross-section, the total surface area of the cone, and its volume.

Solution

To solve this problem, we can break it down into the following steps:

1. Find the radius of the quarter circle formed by the cutting plane. 2. Calculate the area of the cross-section. 3. Calculate the slant height of the cone. 4. Calculate the lateral surface area of the cone. 5. Calculate the total surface area of the cone. 6. Calculate the volume of the cone.

Let's go through each step in detail.

Step 1: Find the radius of the quarter circle

The radius of the quarter circle formed by the cutting plane is equal to the radius of the cone's base, which is given as 3 cm.

Radius of the quarter circle = 3 cm

Step 2: Calculate the area of the cross-section

The area of the cross-section is equal to a quarter of the area of a circle with the given radius.

Area of the cross-section = (1/4) * π * (radius of the quarter circle)^2

Substituting the value of the radius of the quarter circle:

Area of the cross-section = (1/4) * π * (3 cm)^2

Step 3: Calculate the slant height of the cone

To calculate the slant height of the cone, we can use the given angle at the vertex of the cross-section. The slant height can be found using the formula:

Slant height = (radius of the quarter circle) / sin(angle at the vertex)

Substituting the values:

Slant height = (3 cm) / sin(60°)

Step 4: Calculate the lateral surface area of the cone

The lateral surface area of the cone can be calculated using the formula:

Lateral surface area = π * (radius of the cone) * (slant height)

Substituting the values:

Lateral surface area = π * (3 cm) * (slant height)

Step 5: Calculate the total surface area of the cone

The total surface area of the cone includes the base as well. The formula for the total surface area is:

Total surface area = Lateral surface area + Area of the base

The area of the base is given by:

Area of the base = π * (radius of the cone)^2

Substituting the values:

Total surface area = Lateral surface area + π * (3 cm)^2

Step 6: Calculate the volume of the cone

The volume of the cone can be calculated using the formula:

Volume = (1/3) * π * (radius of the cone)^2 * (height of the cone)

Since the height of the cone is not given, we cannot calculate the volume without this information.

Final Answer

To summarize, we have calculated the following:

- Radius of the quarter circle: 3 cm - Area of the cross-section: (1/4) * π * (3 cm)^2 - Slant height of the cone: (3 cm) / sin(60°) - Lateral surface area of the cone: π * (3 cm) * (slant height) - Total surface area of the cone: Lateral surface area + π * (3 cm)^2

Please note that the volume of the cone cannot be calculated without the height of the cone.