Question

Угол развёртки боковой поверхности конуса составляет 60°, а радиус конуса 5 см. Найдите высоту конуса и площадь его полной поверхности .​

Answer 1

Ответ:
h=5√35см
Sпол=175π см²

Решение:
С=2πr=2*5π=10π см длина окружности конуса, и длина дуги боковой развертки (т.е. сектора).
Сдуги=2πR*a/360, где а-градусная мера центрального угла; R- образующая конуса.
а=60°
10π=2πR*60/360
10π=πR/3
R=10*3
R=30см образующая конуса.
Теорема Пифагора
h=√(R²-r²)=√(30²-5²)=√(900-25)=
=√875=5√35см .
Sосн=πr²=5²π=25π см²
Sбок=πrR=5*30π=150πсм²
Sпол=25π+150π=175π см²

Answer 2

Calculation of Cone Height and Total Surface Area

To find the height of the cone and the total surface area, we can use the given information that the angle of the cone's development is 60° and the radius of the cone is 5 cm.

Let's calculate the height of the cone first.

The angle of the cone's development is 60°, which means that the slant height of the cone forms an equilateral triangle with the radius of the cone. In an equilateral triangle, all sides are equal, so the slant height of the cone is also equal to the radius.

Therefore, the slant height of the cone is 5 cm.

To find the height of the cone, we can use the Pythagorean theorem. The height, slant height, and radius of the cone form a right triangle.

Using the Pythagorean theorem, we have:

height^2 + radius^2 = slant height^2

Let's substitute the values:

height^2 + 5^2 = 5^2

Simplifying the equation:

height^2 + 25 = 25

height^2 = 0

Since the height^2 is equal to 0, the height of the cone is 0 cm.

Now, let's calculate the total surface area of the cone.

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

Total Surface Area = π * radius * slant height + π * radius^2

Substituting the values:

Total Surface Area = π * 5 * 5 + π * 5^2

Simplifying the equation:

Total Surface Area = 25π + 25π

Total Surface Area = 50π

Therefore, the total surface area of the cone is 50π square cm.

Please note that the height of the cone is 0 cm, which means that the cone is degenerate and has no volume.