Question

Two steel sheets each of length a₁ and breadth a₂ are used to prepare the surfaces of two right circular cylinders ⇀ one having volume v₁ and height a₂ and other having volume v₂ and height a₁. Then, 

Options

• v₁ = v₂

• a₁v₁ = a₂v₂

•  a₂v₁ = a₁v₂

• \[\frac{v_1}{a_1} = \frac{v_2}{a_2}\]

   

Answer 1

Let the radius of the base of the cylinders be r and R.
Now, let sheet with length a₁ be used to form a cylinder with volume v₁
So,

\[2\pi r = a_1 \]

\[ \Rightarrow r = \frac{a_1}{2\pi}\]

Volume

\[v_1 = \pi r^2 h = \pi \left( \frac{a_1}{2\pi} \right)^2 a_2 = \frac{{a_1}^2 a_2}{4\pi}\]

Similarly, let sheet with length a2 be used to form a cylinder with volume v2
So,

\[2\pi r = a_1 \]

\[ \Rightarrow r = \frac{a_1}{2\pi}\]

Volume 

\[v_2 = \pi R^2 h = \pi \left( \frac{a_2}{2\pi} \right)^2 a_1 = \frac{{a_2}^2 a_1}{4\pi}\]

Now,

\[\frac{v_1}{v_2} = \frac{{a_1}^2 a_2}{{a_2}^2 a_1} = \frac{a_1}{a_2}\]

\[ \Rightarrow v_1 a_2 = v_2 a_1\]