Question

Шар вписан в равносторонний конус с образующей 6 см. Найти объём шара. Спасибо!!!

Answer 1

Радиус вписанного шара=радиус вписанной окружности в равносторонний треугольник, радиус=сторона треугольника*корень3/6=6*корень3/6=корень3, поверхность шара=4пи*радиус в квадрате=4пи*3=12пи

Answer 2

Problem Analysis

We are given that a sphere is inscribed in an equilateral cone with a slant height of 6 cm. We need to find the volume of the sphere.

Solution

To find the volume of the sphere, we need to know its radius. Let's denote the radius of the sphere as r.

Since the sphere is inscribed in the cone, the radius of the sphere is equal to the radius of the cone's base. Let's denote the radius of the cone's base as R.

To find the radius of the cone's base, we can use the slant height and the height of the cone. However, we are not given the height of the cone in this problem.

To proceed, we need to make an assumption about the height of the cone. Let's assume that the height of the cone is equal to the slant height, which is 6 cm.

Now, we can use the slant height and the height of the cone to find the radius of the cone's base using the Pythagorean theorem.

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (in this case, the slant height) is equal to the sum of the squares of the other two sides (in this case, the radius and the height of the cone).

Using the Pythagorean theorem, we can write the equation as:

R^2 = (r^2) + (h^2)

Substituting the values, we have:

R^2 = (r^2) + (6^2)

Simplifying the equation, we have:

R^2 = r^2 + 36

Now, we can find the volume of the sphere using the formula:

Volume of a sphere = (4/3) * π * r^3

Substituting the value of r into the formula, we have:

Volume of the sphere = (4/3) * π * (R^2 - 36)^(3/2)

Let's calculate the volume of the sphere using the given information.

Calculation

Using the formula for the volume of the sphere, we have:

Volume of the sphere = (4/3) * π * (R^2 - 36)^(3/2)

Substituting the value of R^2 from the equation we derived earlier, we have:

Volume of the sphere = (4/3) * π * (r^2 + 36 - 36)^(3/2)

Simplifying the equation, we have:

Volume of the sphere = (4/3) * π * r^3

Therefore, the volume of the sphere is (4/3) * π * r^3.

Answer

The volume of the sphere inscribed in the equilateral cone with a slant height of 6 cm is (4/3) * π * r^3.