Question

A cylindrical gas container is closed at the top and open at the bottom. If the iron plate of the top is $\frac{5}{4}$ times as thick as the plate forming the cylindrical sides, the ratio of the radius to the height of the cylinder using minimum material for the same capacity is :

• A. $\frac{2}{3}$
• B. $\frac{1}{2}$
• C. $\frac{4}{5}$
• D. $\frac{1}{3}$

Answer 1

The correct option is C

$\frac{4}{5}$

The explanation for the correct option:

Step 1: Find the surface area of the cylinder.

Let, $m$represents the thickness of the cylindrical sides.

So, the thickness of the iron plate at the top $=\frac{5m}{4}$

We know that the volume of the cylinder is:

$$\begin{gathered} \Rightarrow V=\pi r^{2}h \\ \Rightarrow h=\frac{V}{\pi r^2} \end{gathered}$$

And the surface area of the cylinder is: $S=2\pi rh+2\pi r^2$

if $m$ be the thickness of the sides then that of the top will be $\frac{5m}{4}$

So the surface area would be:

$$\begin{gathered} \Rightarrow S=\left(2\pi rh\right)m+\left(\pi r^2\right)\frac{5m}{4} \\ \Rightarrow S=(2\pi r\times \frac{V}{\pi r^2}\times m)+\left(\pi r^2\right)\frac{5m}{4} \\ \Rightarrow S=m(\frac{2V}{r}+\frac{5}{4}\pi r^2) \end{gathered}$$

Step 2: Find the first derivative of the surface area of the cylinder:

Now, $\frac{dS}{dr}=0$ So that we get the value of $r^3$ in terms of $V$.

$$\begin{gathered} \Rightarrow \frac{dS}{dr}=\frac{d}{dr}\left(m\right(\frac{2V}{r}+\frac{5}{4}\pi r^2\left)\right) \\ \Rightarrow \frac{dS}{dr}=m\left[2V\frac{d}{dr}\right(\frac{1}{r})+\frac{5}{4}\pi \frac{d}{dr}(r^2\left)\right] \\ \Rightarrow \frac{dS}{dr}=m(\frac{-2V}{r^2}+\frac{5}{2}\pi r) \end{gathered}$$

Step 3: Find the required ratio:

$\frac{dS}{dr}=0$ when, $r^3=\frac{4V}{5\pi }$or $5\pi r^3=4\pi r^{2}h$

$$\begin{gathered} \frac{d^{2}S}{d{r}^2}=m(\frac{4V}{r^3}+\frac{5}{2}\pi ) \\ \frac{d^{2}S}{d{r}^2}>0forr=\frac{4h}{5} \end{gathered}$$

Hence at this value surface area will be minimum

$5\pi r^3=4\pi r^{2}h$

$$\begin{gathered} \Rightarrow \frac{r}{h}=\frac{4\pi }{5\pi } \\ \Rightarrow \frac{\mathbf{r}}{\mathbf{h}}\mathbf{=}\frac{\mathbf{4}}{\mathbf{5}} \end{gathered}$$

Therefore, option (C) is the correct answer.