Question

ПОМОГИТЕ, ОЧЕНЬ СРОЧНО НАДО!!!!!(((((((( Высота конуса равна 5 см. Угол при вершине осевого сечения равен 120º. а) Найти площадь сечения конуса плоскостью, проходящей через две образующие, угол между которыми равен 45º. б) Найти площадь боковой поверхности конуса.

Answer 1

Ответ: а) 25√2 см²; б) 50√3π см²

Объяснение:  

      Пусть АВ - диаметр данного конуса - прямого, поскольку иное не указано. Поэтому его  вершина проецируется в центр О основания. По условию высота МО=5 см, угол АМВ=120°, угол между образующими АМ и СМ равен 45°.

     Осевое сечение конуса - равнобедренный треугольник АМВ,  высота которого ( она же биссектриса) делит на 2 равных прямоугольных треугольника. Угол АМО=120°:2=60°.

=>

Образующая АМ=МС=МО:cos 60°=5:0,5=10 (см).

Сечение между образующими  -  равнобедренный треугольник.

а)

Одна из формул площади треугольника S=0,5•a•b•α , где а и b- стороны, α - угол между ними.

Ѕ( АМС) =АМ•СМ•ѕin45°=0,5•10•10•√2/2=

б)

Формула боковой поверхности конуса S=πRl

R=AM•sin60°=10√3/2=5√3

S=π•(5√3)•10=50√3π см²

Answer 2

Finding the Area of the Cross-Section of the Cone

To find the area of the cross-section of the cone, we need to determine the shape of the cross-section. In this case, the cross-section is formed by a plane passing through two generatrices (the lines connecting the vertex of the cone to points on the base) at an angle of 45º between them.

Let's denote the radius of the base of the cone as r and the height of the cone as h. In this case, the height of the cone is given as 5 cm.

To find the radius of the base, we can use the formula for the slant height of the cone, which is given by:

l = sqrt(r^2 + h^2)

Since the height of the cone is given as 5 cm, we can substitute this value into the formula:

l = sqrt(r^2 + 5^2)

Now, we need to find the value of r. To do this, we can use the given information that the angle at the vertex of the cross-section is 120º. Since the angle between the generatrices is 45º, we can use trigonometric ratios to find the value of r.

Let's consider the right triangle formed by the height of the cone, the radius of the base, and the slant height. The angle opposite the height is 90º, the angle opposite the radius is 45º, and the angle opposite the slant height is 120º.

Using the sine function, we can write:

sin(45º) = r / l

Simplifying this equation, we get:

r = l * sin(45º)

Substituting the value of l from the previous equation, we have:

r = sqrt(r^2 + 5^2) * sin(45º)

Solving this equation will give us the value of r.

Finding the Area of the Lateral Surface of the Cone

To find the area of the lateral surface of the cone, we can use the formula:

A = π * r * l

where A is the area, r is the radius of the base, and l is the slant height of the cone.

Using the value of r obtained from the previous calculation and the given height of the cone, we can substitute these values into the formula to find the area of the lateral surface.

Please note that the exact values of the radius and the area cannot be determined without the specific measurements. However, you can use the provided formulas and calculations to find the values once you have the measurements.

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