Question

Ребята, спасайте! Тему пропустила, как решать не знаю...Через вершину конуса и хорду основания, стягивающую дугу в 120 градусов, проведено сечение, составляющее с плоскостью основания угол в 45 градусов. Найти площадь сечения, если радиус основания равен 4 см.

Answer 1

АВР - сечение, площадь сечения S=1/2 АВ*РС, АВ=2АС, дуга АВ 120°, значит угол АОВ = 120°, тогда угол АОС=60°, АС=АО*sin60°. AO=R=4, AC=4*√3/2=2√3, AB=4√3. Т к сечение, составляет с плоскостью основания угол в 45 градусов, то угол РСО=45°, треугольник РСО прямоугольный равнобедренный, РО=ОС, РС=√2 ОС. Из треугольника АОС ОС=1/2 АО (катет против угла 30°), ОС=2, РС=2√2. S=1/2*4√3*2√2=4√6

Answer 2

To solve this problem, we need to find the area of the cross-section formed by the intersection of the cone and the plane. The cone has a vertex and a base, and the chord of the base forms a 120-degree arc. The cross-section forms a 45-degree angle with the plane of the base.

Given Information:

- Radius of the base of the cone = 4 cm

Solution:

To find the area of the cross-section, we can use the formula for the area of a sector of a circle. The sector is formed by the 120-degree arc of the base.

The formula for the area of a sector is given by:

Area of sector = (θ/360) * π * r^2

Where: - θ is the central angle of the sector in degrees - r is the radius of the base of the cone

In this case, the central angle of the sector is 120 degrees, and the radius of the base is 4 cm. Substituting these values into the formula, we get:

Area of sector = (120/360) * π * (4^2)

Simplifying the equation:

Area of sector = (1/3) * π * 16

Now, we need to find the area of the cross-section formed by the intersection of the cone and the plane. The cross-section forms a 45-degree angle with the plane of the base. Since the cross-section is a right-angled triangle, we can use the formula for the area of a triangle.

The formula for the area of a triangle is given by:

Area of triangle = (1/2) * base * height

In this case, the base of the triangle is the chord of the base, and the height is the distance from the vertex of the cone to the plane of the base. Since the cross-section forms a 45-degree angle with the plane of the base, the height is equal to the radius of the base.

Substituting the values into the formula, we get:

Area of triangle = (1/2) * (chord) * (radius)

To find the chord, we can use the formula for the length of a chord in a circle. The formula is given by:

Chord length = 2 * radius * sin(θ/2)

Where: - θ is the central angle of the sector in radians - radius is the radius of the base of the cone

In this case, the central angle of the sector is 120 degrees, which is equal to 2π/3 radians. Substituting the values into the formula, we get:

Chord length = 2 * 4 * sin(2π/3)

Simplifying the equation:

Chord length = 8 * sin(2π/3)

Now, substituting the values into the formula for the area of the triangle, we get:

Area of triangle = (1/2) * (8 * sin(2π/3)) * 4

Simplifying the equation:

Area of triangle = 16 * sin(2π/3)

To find the area of the cross-section, we need to subtract the area of the triangle from the area of the sector:

Area of cross-section = Area of sector - Area of triangle

Substituting the values into the equation, we get:

Area of cross-section = (1/3) * π * 16 - 16 * sin(2π/3)

Simplifying the equation:

Area of cross-section = (16/3) * (π - sin(2π/3))

Calculating the value, we find:

Area of cross-section ≈ 16.755 cm^2

Therefore, the approximate area of the cross-section formed by the intersection of the cone and the plane is 16.755 square centimeters.