Question

Высота конуса равна 8 а длина образующей 10 см. найти пщадьосевого сечения этого конуса площадьего полной поверхности и объем.

Answer 1

Сделав чертеж, можно увидеть, что образующая, высота, радиус образуют прямоугольный треугольник.
По теореме Пифагора найдем радиус: 10² - 8² = 100 - 64 = 36 = 6², значит, радиус равен 6 см.
Из чертежа также видно, что осевое сечение - равнобедренный треугольник с основанием, равным диаметру (d = 2r), и высотой, равной высоте конуса. Значит Sсеч = 1/2 · 12 · 8 = 48 (см²).
Sполн = Sосн + Sбок = πr² + πrl = πr(r + l) = π · 6 · (6 + 10) = 96π (см²)
V = 1/3 · πr²h = 1/3 · π · 6² · 8 = 96π (см³).

Answer 2

Calculation of Cone Properties

To find the surface area of the base, the total surface area, and the volume of a cone, we need to know the height and the length of the slant height (also known as the generating line) of the cone.

Given: - Height of the cone (h) = 8 cm - Length of the slant height (l) = 10 cm

Surface Area of the Base

The base of a cone is a circle, and its surface area can be calculated using the formula for the area of a circle: A = πr^2, where r is the radius of the base.

To find the radius of the base, we can use the Pythagorean theorem, which states that the square of the hypotenuse (l) is equal to the sum of the squares of the other two sides (r and h). In this case, the hypotenuse is the slant height (l), and the other two sides are the radius (r) and the height (h) of the cone.

Using the Pythagorean theorem, we can calculate the radius (r) as follows:

r = √(l^2 - h^2)

Substituting the given values, we have:

r = √(10^2 - 8^2) = √(100 - 64) = √36 = 6 cm

Now that we have the radius (r), we can calculate the surface area of the base (A) using the formula for the area of a circle:

A = πr^2 = π(6^2) = 36π cm^2 [[1]]

Total Surface Area

The total surface area of a cone includes the area of the base and the curved surface area. The curved surface area can be calculated using the formula: A = πrl, where r is the radius of the base and l is the slant height.

Substituting the given values, we have:

A = π(6)(10) = 60π cm^2 [[2]]

Volume

The volume of a cone can be calculated using the formula: V = (1/3)πr^2h, where r is the radius of the base and h is the height.

Substituting the given values, we have:

V = (1/3)π(6^2)(8) = 96π cm^3 [[3]]

Summary

Based on the given information, the properties of the cone are as follows: - Surface area of the base: 36π cm^2 - Total surface area: 60π cm^2 - Volume: 96π cm^3

Please let me know if there is anything else I can help you with!