Question

ПОЖАЛУЙСТА!!!ПОМОГИТЕ!!!! Образующая конуса наклонена к плоскости основания под углом 60 градусов.Радиус основания конуса равен 6.Найдите объем конуса и площадь его боковой поверхности

Answer 1

из вершины конуса надо опустить перпендикуляр на основание. пусть это будет AH.точка H как раз будет находиться на диаметре окружности(основания).пусть диаметр будет MN. следовательно,мы получим прямоугольный треугольник AHN,в котором угол ANH=60 градусов. r=HN=cos60*40=20(т.к. HN-прилежащий катет,а AN- гипотенуза ---> cos60=HN/AN,где AN-образующая).

S=Pi*r*l=Pi*AN*HN=3.14*20*40=2512 

Answer 2

Calculation of Cone Volume and Surface Area

To find the volume and surface area of a cone, we need to use the given information: the inclination angle of the cone's generatrix to the base plane, which is 60 degrees, and the radius of the cone's base, which is 6.

Volume of the Cone

The formula to calculate the volume of a cone is:

V = (1/3) * π * r^2 * h

Where: - V is the volume of the cone - π is a mathematical constant approximately equal to 3.14159 - r is the radius of the cone's base - h is the height of the cone

In this case, we need to find the height of the cone. To do that, we can use trigonometry. The height of the cone is the distance from the apex to the base along the generatrix. Since the generatrix is inclined at an angle of 60 degrees to the base plane, we can use the sine function to find the height.

sin(60) = h / r

Rearranging the equation, we get:

h = r * sin(60)

Substituting the given values, we have:

h = 6 * sin(60)

Now we can calculate the volume of the cone using the formula mentioned earlier:

V = (1/3) * π * r^2 * h

Substituting the values of r and h, we get:

V = (1/3) * π * 6^2 * (6 * sin(60))

Simplifying the expression, we have:

V = (1/3) * π * 36 * (6 * 0.866)

V = (1/3) * π * 36 * 5.196

V ≈ 120.48 cubic units

Therefore, the volume of the cone is approximately 120.48 cubic units.

Surface Area of the Cone

The formula to calculate the surface area of a cone is:

A = π * r * (r + l)

Where: - A is the surface area of the cone - π is a mathematical constant approximately equal to 3.14159 - r is the radius of the cone's base - l is the slant height of the cone

To find the slant height, we can use the Pythagorean theorem. The slant height is the distance from the apex to any point on the circumference of the base. It forms a right triangle with the height and the radius of the base.

Using the Pythagorean theorem, we have:

l^2 = h^2 + r^2

Substituting the values of h and r, we get:

l^2 = (6 * sin(60))^2 + 6^2

l^2 = (6 * 0.866)^2 + 6^2

l^2 = 36 * 0.75 + 36

l^2 = 27 + 36

l^2 = 63

l ≈ √63

Now we can calculate the surface area of the cone using the formula mentioned earlier:

A = π * r * (r + l)

Substituting the values of r and l, we get:

A = π * 6 * (6 + √63)

Simplifying the expression, we have:

A ≈ π * 6 * (6 + √63)

A ≈ 6π * (6 + √63)

A ≈ 6 * 3.14159 * (6 + √63)

A ≈ 113.1 square units

Therefore, the surface area of the cone is approximately 113.1 square units.

Conclusion

The volume of the cone is approximately 120.48 cubic units, and the surface area of its lateral surface is approximately 113.1 square units.