Question

В конусе проведена плоскость, параллельная оси и отсекающая от окружности основания дугу, равную 120°. Высота конуса 12 см, расстояние от секущей плоскости до оси 3 см. Найдите площадь сечения. (с рисунком)

Answer 1

Ответ:

54 (см²).

Объяснение:

рассмотрите предложенное решение; по возможности перепроверьте расчёты.

Answer 2

Finding the Area of the Cross-Section of a Cone

To find the area of the cross-section of a cone, we need to determine the shape of the cross-section. In this case, the cross-section is a circle with a central angle of 120°.

Given information: - Cone height: 12 cm - Distance from the cutting plane to the axis: 3 cm

To find the radius of the circle, we can use the properties of similar triangles. The triangle formed by the height of the cone, the distance from the cutting plane to the axis, and the radius of the circle is similar to the triangle formed by the height of the cone, the slant height of the cone, and the radius of the cone.

Using the given information, we can set up the following proportion:

height of cone / distance from cutting plane to axis = slant height of cone / radius of cone

Substituting the given values: 12 cm / 3 cm = slant height of cone / radius of cone

Simplifying the proportion: 4 = slant height of cone / radius of cone

Since the slant height of the cone is always greater than the height, we can conclude that the radius of the cone is 1/4 of the slant height.

Now, let's find the slant height of the cone. Using the Pythagorean theorem, we can set up the following equation:

slant height of cone^2 = height of cone^2 + radius of cone^2

Substituting the given values: slant height of cone^2 = 12 cm^2 + (1/4 * slant height of cone)^2

Simplifying the equation: slant height of cone^2 = 144 cm^2 + (1/16 * slant height of cone)^2

Multiplying both sides by 16 to eliminate the fraction: 16 * slant height of cone^2 = 2304 cm^2 + slant height of cone^2

Simplifying further: 15 * slant height of cone^2 = 2304 cm^2

Dividing both sides by 15: slant height of cone^2 = 153.6 cm^2

Taking the square root of both sides: slant height of cone ≈ 12.4 cm

Since the radius of the cone is 1/4 of the slant height, the radius of the cone is approximately 3.1 cm.

Now that we have the radius of the cone, we can find the area of the cross-section. The area of a circle is given by the formula:

Area = π * radius^2

Substituting the value of the radius: Area = π * (3.1 cm)^2

Calculating the area: Area ≈ 30.2 cm^2

Therefore, the area of the cross-section of the cone is approximately 30.2 square centimeters.


Please note that the diagram provided is for illustrative purposes only and may not be to scale.