Question

Через вершину конуса проведена плоскость пересекающая основание по хорде, длина которой равна 6. Угол между образующими в сечении равен 90˚, а наибольший угол между образующими конуса равен 120 ˚. Найдите площадь полной поверхности конуса. (С рисунком плз)

Answer 1

Рисунки 1 - вид сверху, 2 - вид на плоскость сечения с хордой 6, 3 - осевое сечение конуса.

Answer 2

Finding the Surface Area of a Cone

To find the surface area of a cone, we need to know the length of the chord on the base and the angle between the generatrices in the section.

Given: - Length of the chord on the base: 6 - Angle between the generatrices in the section: 90° - Largest angle between the generatrices of the cone: 120°

To calculate the surface area of the cone, we can use the following formula:

Surface Area = π * r * (r + l)

Where: - r is the radius of the base of the cone - l is the slant height of the cone

To find the radius of the base, we can use the length of the chord on the base. The formula to calculate the radius from the chord length is:

r = (l^2 + c^2) / (2 * l)

Where: - c is the length of the chord on the base - l is the slant height of the cone

To find the slant height of the cone, we can use the angle between the generatrices in the section. The formula to calculate the slant height from the angle is:

l = c / (2 * sin(θ/2))

Where: - c is the length of the chord on the base - θ is the angle between the generatrices in the section

Let's calculate the surface area of the cone step by step.

Step 1: Calculate the Slant Height

Using the formula for the slant height, we can calculate:

l = 6 / (2 * sin(90°/2))

Calculating the value, we get:

l = 6 / (2 * sin(45°))

l ≈ 6 / (2 * 0.7071) ≈ 4.2426

So, the slant height of the cone is approximately 4.2426.

Step 2: Calculate the Radius

Using the formula for the radius, we can calculate:

r = (4.2426^2 + 6^2) / (2 * 4.2426)

Calculating the value, we get:

r = (18 + 36) / 8.4852 ≈ 5.6569

So, the radius of the base of the cone is approximately 5.6569.

Step 3: Calculate the Surface Area

Using the formula for the surface area, we can calculate:

Surface Area = π * 5.6569 * (5.6569 + 4.2426)

Calculating the value, we get:

Surface Area ≈ 3.1416 * 5.6569 * 9.8995 ≈ 177.715

So, the approximate surface area of the cone is 177.715 square units.

Please note that the calculations are approximate due to rounding.

Here is a diagram to help visualize the cone:

          /|
         / |
        /  |
       /   |
      /    |
     /     |
    /      |
   /       |
  /        |
 /_________|

I hope this helps! Let me know if you have any further questions.