Amongst all open (from the top) right circular cylindrical boxes of volume 125\pi cm^{3}, find the dimensions of the box which has the least surface area.

 

Answers (1)

Let radius of the cylinder - r

height of the cylinder - h

\\ $Volume of the cylinder $ \\ V = \pi \mathrm{r}^{2} \mathrm{h}=125 \pi

\\ \mathrm{h}=\frac{125}{\mathrm{r}^{2}}

\\ $ Surface Area of this cylinderical boxex $\\ \mathrm{S}=2 \pi \mathrm{rh}+\pi \mathrm{r}^{2} \\

\mathrm{S}=\frac{250 \pi}{\mathrm{r}}+\pi \mathrm{r}^{2}

\\ \frac{\mathrm{d} \mathrm{S}}{\mathrm{dr}}=\frac{-250 \pi}{\mathrm{r}^{2}}+2 \pi \mathrm{r}

For critical point 

\\ \frac{\mathrm{dS}}{\mathrm{dr}}=0 \\ \\ \frac{-250 \pi}{\mathrm{r}^{2}}+2 \pi \mathrm{r} = 0 \\\\ \mathrm{r}^3=125 \ \ \ \Rightarrow \mathrm{r}=5\\

For maximum and minimum valle

\frac{\mathrm{d}^{2} \mathrm{S}}{\mathrm{dr}^{2}}=\frac{500 \pi}{\mathrm{r}^{3}}+2 \pi>0

Surface area is minimum at r = 5cm and \\ \mathrm{h}=\frac{125}{\mathrm{5}^{2}}=5 \ cm.

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