Question

диагональ осевого сечения цилиндра равна 48 см угол между этой диагональю и образующей цилиндра 60 градусов Найдите площадь полной поверхности цилиндра

Answer 1

Ответ:

Объяснение:

Осевое сечение цилиндра прямоугольник с диагональю LM=48 см.

Она является гипотенузой и равна двум образующим,так как образующая лежит против угла в 30°

KL=1/2LM=1/2×48=24 см

KM=√LM²-KL²=√48²-24²=√1728=24√3  см

KM=d

r=1/2d=1/2×24√3=12√3  см

Sосн.=πr²=(12√3)²π=432π  см²

Sбоk.=2πrKL=2×π×12√3 ×24=576√3π  см²

Sполн.=2Sосн.+Sбоk.=2×432π +576√3π  =1440√3π см²

Answer 2

Calculation of the Surface Area of a Cylinder

To find the surface area of a cylinder, we need to consider two components: the curved surface area and the area of the two circular bases.

Let's start by calculating the curved surface area of the cylinder.

The curved surface area of a cylinder can be calculated using the formula:

Curved Surface Area = 2 * π * r * h

Where: - π is a mathematical constant approximately equal to 3.14159. - r is the radius of the cylinder. - h is the height of the cylinder.

In this case, we are given the diagonal of the cross-section of the cylinder, which is equal to 48 cm, and the angle between this diagonal and the generator of the cylinder is 60 degrees.

To find the radius and height of the cylinder, we can use trigonometric relationships.

Let's consider the right triangle formed by the diagonal, the radius, and the height of the cylinder. The diagonal is the hypotenuse, the radius is the adjacent side, and the height is the opposite side.

Using trigonometric functions, we can relate the given angle and the sides of the triangle:

tan(angle) = opposite / adjacent

In this case, the angle is 60 degrees, the opposite side is the height, and the adjacent side is the radius.

Let's solve for the height first:

tan(60) = height / radius

Simplifying the equation:

height = radius * tan(60)

Now, we can use the Pythagorean theorem to relate the diagonal, the radius, and the height:

diagonal^2 = radius^2 + height^2

Substituting the value of the height:

48^2 = radius^2 + (radius * tan(60))^2

Simplifying the equation:

48^2 = radius^2 + (radius^2 * tan(60)^2)

Simplifying further:

48^2 = radius^2 * (1 + tan(60)^2)

Solving for the radius:

radius = sqrt(48^2 / (1 + tan(60)^2))

Now that we have the radius, we can calculate the height using the equation:

height = radius * tan(60)

Once we have the radius and height, we can calculate the curved surface area using the formula mentioned earlier:

Curved Surface Area = 2 * π * radius * height

Let's calculate the values and find the surface area of the cylinder.