Question

Радиус основания цилиндра равен 6 см, а диагональ его осевого сечения образует с плоскостью основания 60°. Найти: 1. высоту цилиндра2. площадь осевого сечения цилиндра 3. площадь поверхности цилиндрасрочно надо, помогите ​

Answer 1

Пошаговое объяснение:

1. Высота цилиндра - это один из катетов треугольника, образованного диаметром и диагональю сечения. D = 2r = 12.

В треугольнике  угол = 60, прилежащий катет 12. Противолежащий: 12*tg(pi/3) = 12√3

2. Площадь осевого сечения складывается из двух таких треугольников. Оба катета известны, находим площадь:

12*12√3/2 * 2 = 144√3

3. S = 2pi*r*h + 2pi*r^2

2*3,14*6*12√3 + 2*3,14*36 = 75,36√3 + 226,08

Answer 2

Problem Analysis

To solve this problem, we need to find the height of the cylinder, the area of its axial section, and the surface area of the cylinder. We are given the radius of the base of the cylinder (6 cm) and the angle between the diagonal of the axial section and the plane of the base (60°).

1. Finding the Height of the Cylinder

To find the height of the cylinder, we can use trigonometry. The diagonal of the axial section forms a right triangle with the height of the cylinder. The angle between the diagonal and the base is 60°, and the radius of the base is given as 6 cm.

Let's denote the height of the cylinder as h. In the right triangle, the opposite side is h and the hypotenuse is twice the radius of the base (12 cm). We can use the sine function to find the height:

sin(60°) = h / 12

Simplifying the equation, we have:

h = 12 * sin(60°)

Now, let's calculate the value of h.

2. Finding the Area of the Axial Section

The area of the axial section of the cylinder can be calculated using the formula for the area of an equilateral triangle. Since the angle between the diagonal and the base is 60°, the axial section is an equilateral triangle.

The formula for the area of an equilateral triangle is:

Area = (sqrt(3) / 4) * side^2

In this case, the side of the equilateral triangle is equal to the diameter of the base, which is twice the radius (12 cm). Let's calculate the area of the axial section.

3. Finding the Surface Area of the Cylinder

The surface area of the cylinder can be calculated by summing the areas of the two bases and the lateral surface area.

The area of each base is given by the formula:

Base Area = π * radius^2

The lateral surface area is given by the formula:

Lateral Surface Area = 2 * π * radius * height

Let's calculate the surface area of the cylinder.

Solution

1. Finding the height of the cylinder: - Using the equation h = 12 * sin(60°), we can calculate the height of the cylinder. - Substituting the value of sin(60°) as 0.866, we have: h = 12 * 0.866 = 10.392 cm

2. Finding the area of the axial section: - Using the formula for the area of an equilateral triangle, we can calculate the area of the axial section. - Substituting the value of the side as 12 cm, we have: Area = (sqrt(3) / 4) * 12^2 = 36 * sqrt(3) cm^2

3. Finding the surface area of the cylinder: - The base area is given by Base Area = π * radius^2. - The lateral surface area is given by Lateral Surface Area = 2 * π * radius * height. - Substituting the values of the radius (6 cm) and height (10.392 cm), we have: Base Area = π * 6^2 = 36π cm^2 Lateral Surface Area = 2 * π * 6 * 10.392 = 124.704π cm^2 - The total surface area is the sum of the base area and the lateral surface area: Surface Area = Base Area + Lateral Surface Area = 36π + 124.704π = 160.704π cm^2

Answer

1. The height of the cylinder is 10.392 cm. 2. The area of the axial section of the cylinder is 36 * sqrt(3) cm^2. 3. The surface area of the cylinder is 160.704π cm^2.