Question

The volume of greatest cone inacribed in a cube of side a is

Answer 1

Answer : $\frac{\pi}{12} {a}^{3}$

Step-by-step explanation :

If a cone is inscribed in a cube of side a, then,

Diameter of the cone = a.

Height of the cone = a

We know that, 2 × Radius = Diameter.

So,

Radius = a/2

We know that,

Volume of the cone = $\frac{1}{ 3} \pi {r}^{2} h$

Now,

$\sf \: Volume \: of \: the \: cone \: \\ = \frac{1}{3} \pi ({ \frac{a}{2} }^{2} )(a) \\ = \frac{1}{3} \times \frac{1}{4} \pi {a}^{3} \\ = \frac{1}{12} \pi {a}^{3}$

Therefore, Volume of the greatest cone that can be inscribed in a cube of side a is $\frac{\pi}{12} {a}^{3}$

Answer 2

Answer :

The volume of the greatest cone inscribed in a cube of side a is $\large{\frac{\pi}{12}a^3}$ .

Step-by-step explanation :

Let us assume that a cone is inscribed in a cube of side a.

Then, the value of

Diameter of the cone = a

Height of the cone = a

Also, Radius = Diameter ÷ 2

Therefore, the new radius -

⇒ a / 2

Since, we know that

Volume of the cone = $\large{\frac{1}{3}\pi r^2h}$

Now,

Volume of the cone,

⇒ $\large{\frac{1}{3}\pi (\frac{a^2}{2})(a)}$

⇒ $\large{\frac{1}{3}\times\frac{1}{4}\pi a^3}$

⇒ $\large{\frac{1}{12}\pi a^3}$

⇒ $\large{\frac{\pi}{12}a^3}$