Question

В цилиндре диагональ осевого сечения 10см, а диаметр основания 6см. Найти полную поверхность и объём цилиндра.

Answer 1

1. Находим высоту

$h= \sqrt{10^2-6^2} =8$

2. Радиус основания

r=d/2=6/2=3

3. Площадь полной поверхности

S = 2πr (r+h)=2π*3*(3+8)=66π

4. Обьём 

V = π*r²*h=π*3²*8=72π

Answer 2

Calculation of the Cylinder's Surface Area and Volume

To find the surface area and volume of a cylinder, we need to use the given measurements of the cylinder's diagonal of the axial section and the diameter of the base.

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

Surface Area:

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

Surface Area = 2πr(r + h)

where: - r is the radius of the base, which is half the diameter of the base, and - h is the height of the cylinder.

In this case, the diameter of the base is given as 6 cm, so the radius is r = 6 cm / 2 = 3 cm.

To find the height of the cylinder, we can use the Pythagorean theorem. The diagonal of the axial section is given as 10 cm, which represents the hypotenuse of a right triangle formed by the radius, height, and diagonal. The radius is one of the legs of the triangle.

Using the Pythagorean theorem, we can calculate the height:

height = √(diagonal^2 - radius^2)

Substituting the given values, we get:

height = √(10 cm^2 - 3 cm^2) = √(100 cm^2 - 9 cm^2) = √91 cm ≈ 9.54 cm

Now we can substitute the values of the radius and height into the surface area formula:

Surface Area = 2π(3 cm)(3 cm + 9.54 cm) ≈ 2π(3 cm)(12.54 cm) ≈ 75.4 cm^2

Therefore, the full surface area of the cylinder is approximately 75.4 cm^2.

Volume:

The volume of a cylinder can be calculated using the formula:

Volume = πr^2h

Substituting the values of the radius and height we calculated earlier:

Volume = π(3 cm)^2(9.54 cm) ≈ 85.7 cm^3

Therefore, the volume of the cylinder is approximately 85.7 cm^3.

To summarize: - The full surface area of the cylinder is approximately 75.4 cm^2. - The volume of the cylinder is approximately 85.7 cm^3.

Please note that these calculations are based on the given measurements and assumptions.