Question

Диагональ осевого сечения цилиндра равна 8см и составляет с образующей угол 60градусов. найти площадь полной поверхности цилиндра

Answer 1

Диаметр основания
D = L*cos 60 = 8*0.5 = 4 см -  диаметр
Высота цилиндра
H = L * sin 60 = 8*(√3/2) = 4√3
Боковая поверхность цилиндра
Sбок = π*D*H = π*4*4*√3 = 16√3*π
Площадь оснований (двух)
Sосн = 2* π*D²/4 = 2*π*4²/4 = 8*π
ОТВЕТ S=(16√3 + 8)*π - полная поверхность
Рисунок к задаче - в приложении.

Answer 2

Calculation of the Surface Area of a Cylinder

To find the surface area of a cylinder, we need to calculate the sum of the areas of its two circular bases and the lateral surface area.

Given: - The diagonal of the base of the cylinder is 8 cm. - The angle between the diagonal and the height of the cylinder is 60 degrees.

To calculate the surface area, we need to find the radius of the base of the cylinder. Let's denote the radius as r.

To find the radius, we can use the formula for the diagonal of a right circular cylinder:

diagonal = 2r

From this equation, we can solve for r:

r = diagonal / 2

Substituting the given value of the diagonal (8 cm) into the equation, we get:

r = 8 cm / 2 = 4 cm

Now that we have the radius, we can calculate the surface area of the cylinder.

Calculation of the Surface Area of the Bases

The surface area of a circle is given by the formula:

Area = πr^2

Since the cylinder has two bases, we need to calculate the area of each base and then sum them.

The area of one base is:

Area_base = πr^2

Substituting the value of r (4 cm) into the equation, we get:

Area_base = π(4 cm)^2

Calculating the value, we find:

Area_base = 16π cm^2

Since there are two bases, the total area of the bases is:

Total_area_bases = 2 * Area_base

Substituting the value of Area_base, we get:

Total_area_bases = 2 * 16π cm^2

Calculating the value, we find:

Total_area_bases = 32π cm^2

Calculation of the Lateral Surface Area

The lateral surface area of a cylinder is given by the formula:

Lateral_area = 2πrh

where h is the height of the cylinder.

To find the height, we can use the formula for the diagonal and the height of a right circular cylinder:

diagonal = √(r^2 + h^2)

From this equation, we can solve for h:

h = √(diagonal^2 - r^2)

Substituting the given values of the diagonal (8 cm) and the radius (4 cm) into the equation, we get:

h = √((8 cm)^2 - (4 cm)^2)

Calculating the value, we find:

h = √(64 cm^2 - 16 cm^2) = √48 cm^2 = 4√3 cm

Now that we have the height, we can calculate the lateral surface area:

Lateral_area = 2πrh

Substituting the values of r (4 cm) and h (4√3 cm) into the equation, we get:

Lateral_area = 2π(4 cm)(4√3 cm)

Calculating the value, we find:

Lateral_area = 32π√3 cm^2

Calculation of the Total Surface Area

The total surface area of the cylinder is the sum of the areas of the bases and the lateral surface area:

Total_area = Total_area_bases + Lateral_area

Substituting the values of Total_area_bases (32π cm^2) and Lateral_area (32π√3 cm^2) into the equation, we get:

Total_area = 32π cm^2 + 32π√3 cm^2

Calculating the value, we find:

Total_area = 32π(1 + √3) cm^2

Therefore, the total surface area of the cylinder is 32π(1 + √3) cm^2.

Please note that the final answer is an exact value and cannot be simplified further.