Question

Конус получается при вращении прямоугольного треугольника с катетами 6 и 8 вокруг большого катета.Найти объем и площадь поверхности

Answer 1

V равен 1.3*площадь основания*высоту конуса, больший катет равный восьми и есть высота, 6 - радиус основания конуса.
V=1/3*пи*36*8=96 пи
Площадь боковой поверхности: равна <пиrl>. Найдем образующую (она является гипотенузой в прямоугольном треугольнике) l=√(36+64)=10
S=пи*6*10=60пи
площедь полной поверхности(если нужна): пиrl+пиr^2=60пи+36пи=96пи

Answer 2

Finding the Volume and Surface Area of a Cone

To find the volume and surface area of a cone, we need to know the dimensions of the cone. In this case, the cone is formed by rotating a right-angled triangle with legs measuring 6 and 8 units around the longer leg.

Let's calculate the volume and surface area of the cone step by step.

Calculating the Volume of the Cone

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

Volume = (1/3) * π * r^2 * h

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

In this case, the base of the cone is a circle with a radius equal to half the length of the longer leg of the right-angled triangle, which is 8 units. The height of the cone is equal to the length of the shorter leg of the right-angled triangle, which is 6 units.

Let's substitute the values into the formula to calculate the volume:

Volume = (1/3) * π * (8/2)^2 * 6

Simplifying the equation:

Volume = (1/3) * π * 4^2 * 6

Volume = (1/3) * π * 16 * 6

Volume = (1/3) * π * 96

Using the approximate value of π as 3.14159:

Volume ≈ (1/3) * 3.14159 * 96

Volume ≈ 100.5309648

Therefore, the volume of the cone is approximately 100.53 cubic units.

Calculating the Surface Area of the Cone

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

Surface Area = π * r * (r + l)

Where: - π is a mathematical constant approximately equal to 3.14159 - r is the radius of the base of the cone - l is the slant height of the cone

To calculate the slant height, we can use the Pythagorean theorem:

l = √(r^2 + h^2)

In this case, the radius of the base of the cone is 8/2 = 4 units, and the height of the cone is 6 units. Let's calculate the slant height:

l = √(4^2 + 6^2)

l = √(16 + 36)

l = √52

Using the approximate value of √52 as 7.2111:

l ≈ 7.2111

Now, let's substitute the values into the surface area formula:

Surface Area = π * 4 * (4 + 7.2111)

Surface Area = π * 4 * 11.2111

Using the approximate value of π as 3.14159:

Surface Area ≈ 3.14159 * 4 * 11.2111

Surface Area ≈ 141.3716696

Therefore, the surface area of the cone is approximately 141.37 square units.

To summarize: - The volume of the cone is approximately 100.53 cubic units. - The surface area of the cone is approximately 141.37 square units.

Please note that these calculations are approximate due to the use of an approximate value for π.