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,

(a) v₁ = v₂

(b) a₁v₁ = a₂v₂

(c) a₂v₁ = a₁v₂

(d) $\frac{v_1}{a_1}=\frac{v_2}{a_2}$

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,
$$\begin{gathered} 2\mathrm{\pi r}={\mathrm{a}}_1 \\ \Rightarrow \mathrm{r}=\frac{{\mathrm{a}}_1}{2\mathrm{\pi }} \end{gathered}$$
Volume $v_1={\mathrm{\pi r}}^2\mathrm{h}=\mathrm{\pi }{\left(\frac{{\mathrm{a}}_1}{2\mathrm{\pi }}\right)}^2{\mathrm{a}}_2=\frac{{{\mathrm{a}}_1}^2{\mathrm{a}}_2}{4\mathrm{\pi }}$
Similarly, let sheet with length a2 be used to form a cylinder with volume v2
So,
$$\begin{gathered} 2\pi R={\mathrm{a}}_2 \\ \Rightarrow R=\frac{{\mathrm{a}}_2}{2\mathrm{\pi }} \end{gathered}$$
Volume $v_2=\pi R^2\mathrm{h}=\mathrm{\pi }{\left(\frac{{\mathrm{a}}_2}{2\mathrm{\pi }}\right)}^2{\mathrm{a}}_1=\frac{{{\mathrm{a}}_2}^2{\mathrm{a}}_1}{4\mathrm{\pi }}$
Now,
$$\begin{gathered} \frac{v_1}{v_2}=\frac{{a_1}^2{a}_2}{{a_2}^2{a}_1}=\frac{a_1}{a_2} \\ \Rightarrow v_1{a}_2=v_2{a}_1 \end{gathered}$$
Hence, the correct option is (c).