Question

A cylinder and cone have bases of equal radii and are of equal hights. show that thier volumes are in the ratio of 3:1​

Answer 1

Given :–

• A cylinder and cone have bases of equal radii and heights.

To Prove :–

• That their volumes are in the ratio of 3:1

Proof :–

Let the radius be r.

And the height be h.

So,

We know that,

• Volume of cylinder = πr²h

And we also know that,

• Value of cone = $\dfrac{1}{3}$πr²h

So,

We need to prove that 3:1 is the ratio of the volume of cylinder and the volume of cone.

So,

Ratio = Volume of cylinder : Volume of cone

⟹ $\cancel{\pi}$ r²h : $\dfrac{1}{3}$ $\cancel{\pi}$ r²h

The L.H.S. pi (π) cancel with the R.H.S. pi (π), we obtain

⟹ $\cancel{{r}^{2}}$ h : $\dfrac{1}{3}$$\cancel{{r}^{2}}$h

Now, the L.H.S. r² cancel with the R.H.S. r², we obtain

⟹ $\cancel{h}$ : $\dfrac{1}{3}$ $\cancel{h}$

Now, the L.H.S. h cancel with the R.H.S. h, we obtain

⟹ $\dfrac{1}{3}$

So, it means,

⟹ 3:1

Hence,

The ratio of their volumes are 3:1.

★ Hence Proved ★

Answer 2

Ratio = 3 : 1

Given :-

• A cylinder and cone have bases of equal radii and are of equal hights.

To Find :-

• Show that thier volumes are in the ratio of 3:1 .

Solution :-

As we know that,

• Volume of cylinder = πr²h
• Volume of cone = ⅓πr²h

ATQ,

=> Ratio = Volume of cylinder : Volume of cone

=> Ratio = πr²h : ⅓πr²h

=> Ratio = ⅓

=> Ratio = 3 : 1

Thus, The ratio of their volumes are 3 : 1.