Question

Найти объем куба,описанного около конуса , если объем конуса равен 4/3пи

Answer 1

Основание конуса - окружность, вписанная в квадрат-грань куба. 
Высота конуса равна высоте куба. Радиус окружности равен половине ребра куба. Пусть ребро куба x см.$V_{KOH}=\frac13\cdot\pi r^2H=\frac13\pi\left(\frac x2\right)^2x=\frac13\pi\frac14x^3=\frac1{12}\pi x^3\\\frac1{12}\pi x^3=\frac43\pi\\x^3=V_{Ky6}=16$

Answer 2

Конус вписан в куб.
основание конуса- круг вписан в основание куба-квадрат. =>
a=2R
Hконуса=Hкуба=a=2R

Vконуса=(1/3)πR² *H
V=(1/3)πR² * (2R), V=(2/3)πR³
4/3π=2/3πR³
R³=2

Vкуба=a³, V=(2R)³, V=8R³

Vкуба=8*2
Vкуба=16

Answer 3

Calculating the Volume of a Cube Inscribed in a Cone

To find the volume of a cube inscribed in a cone, we can use the formula for the volume of a cone and the properties of the cube.

The volume of a cone is given by the formula: $$V=\frac{\pi R^2H}{3}$$ where $R$ is the radius of the base of the cone and $H$ is the height of the cone.

The volume of a cube is given by the formula: $$V_{\text{cube}}=s^3$$ where $s$ is the length of the side of the cube.

To find the volume of the cube inscribed in the cone, we need to determine the relationship between the cone and the inscribed cube.

Relationship between the Cone and the Inscribed Cube

In this case, the cube is inscribed in the cone, meaning that the vertices of the cube touch the surface of the cone. The cube's vertices touch the circular base of the cone, and its edges touch the lateral surface of the cone.

Let's denote the side length of the cube as $s$. The diagonal of the base of the cube is equal to the diameter of the base of the cone, which is $2R$. Therefore, we have: $$s\sqrt{2}=2R$$ Solving for $s$, we get: $$s=\frac{2R}{\sqrt{2}}=R\sqrt{2}$$

Calculating the Volume of the Inscribed Cube

Now that we have the relationship between the side length of the cube and the radius of the cone, we can calculate the volume of the cube.

The volume of the cube inscribed in the cone is: $$V_{\text{cube}}=s^3=(R\sqrt{2})^3=2R^3$$

Conclusion

Therefore, the volume of the cube inscribed in the cone is twice the volume of the cone, which is $2 \times \frac{4}{3}\pi R^3 = \frac{8}{3}\pi R^3$.

So, if the volume of the cone is $\frac{4}{3}\pi$, then the volume of the cube inscribed in the cone is $\frac{8}{3}\pi$.

This calculation shows that the volume of the cube inscribed in the cone is twice the volume of the cone.