Question

A blacksmith has a rectangular sheet of iron. He has to make a cylindrical vessel both circular ends are closed. When he minimises the wastage of the sheet of iron, then what is the ratio of the wastage to the utilised area of the sheet?

• A. $\frac{1}{11}$
• B. $\frac{2}{17}$
• C. $\frac{3}{22}$
• D. None of these

Answer 1

The correct option is A $\frac{1}{11}$

For the minimum wastage of sheet, he has to cut the sheet is the given manner.


Total area of sheet required = $(2\pi r+4r)\times 2r=4r^2(\pi +2)$
Area of sheet utilised $=(2\pi r\times 2r)+2\left(\pi r^2\right)=6\pi r^2$
Area of wastage sheet $=4r^2(\pi +2)-6\pi r^2=8r^2-2\pi r^2$
$\therefore$ Required ratio $=\frac{8r^2-2\pi r^2}{6\pi r^2}=\frac{2r^2(4-\pi )}{6r^2\pi }=\frac{1}{11}$