Find the minimum length in cm and correct to nearest whole number of the thin metal sheet required to make a hollow and closed cylindrical box of diameter 20 cm and height 35 cm. Given that the width of the metal sheet is 1 m. Also, find the cost of the sheet at the rate of Rs. 56 per m. Find the area of metal sheet required, if 10% of it is wasted in cutting, overlapping, etc.
Find the minimum length in cm and correct to nearest whole number of the thin metal sheet required to make a hollow and closed cylindrical box of diameter 20 cm and height 35 cm. Given that the width of the metal sheet is 1 m. Also, find the cost of the sheet at the rate of Rs. 56 per m. Find the area of metal sheet required, if 10% of it is wasted in cutting, overlapping, etc.

Height of the cylinder box = h = 35 cm

Base radius of the cylinder box = r = 10 cm

Width of metal sheet = 1m = 100 cm

Area of metal sheet required = total surface area of the box

Length x width = 2πr(r + h)

Length x 100 = 2 x 22/7 x 10(10 + 35)

Length x 100 = 22 x 22/7 x 10 x 45

Length = (2 x 22 x 10 x 45)/ (100 x 7) = 28.28 cm = 28 cm

Area of metal sheet = length x width

    \[=\text{ }28\text{ }x\text{ }100\text{ }=\text{ }2800\text{ }c{{m}^{2}}~=\text{ }0.28\text{ }{{m}^{2}}\]

So, the cost of the sheet at the rate of Rs 56 per m2 = Rs (56 x 0.28) = Rs 15.68

Let the total sheet required be x.

    \[\begin{array}{*{35}{l}} x\text{ }-\text{ }10%\text{ }of\text{ }x\text{ }=\text{ }2800\text{ }c{{m}^{2}}  \\ x\text{ }-\text{ }10/100\text{ }\times \text{ }x\text{ }=\text{ }2800\text{ }c{{m}^{2}}  \\ 9x\text{ }=\text{ }2800  \\ x\text{ }=\text{ }3111\text{ }c{{m}^{2}}  \\ \end{array}\]

Therefore, a metal sheet of area 3111 cmis required.