Question

A thin rectangular sheet of metal is 6 inches wide and 10 inches long. The sheet of metal is to be rolled to form a cylinder so that one dimension becomes the circumference of the cylinder and the other dimension becomes the height. What is the volume of the largest possible cylinder formed

Answer 1

Answer:

Length of the Rectangle is to be considered as the circumference of the resulting cylinder and the width of the rectangle should be considered as the height of the cylinder only then we get the maximum volume of the cylinder which is 47.77$in^3$

Step-by-step explanation:

The given dimensions are

Width of the rectangle is  6inches.

Length of the rectangle 10inches.

Now we need to choose in what orientation we need to use the length and the width of the rectangle so that it can be rolled into a rectangle and we get the maximum volume of the resulting cylinder.

so lets consider two cases

CASE-1: width of the rectangle is to be rolled as the circumference of the cylinder and length of the rectangle is to used as the height of the cylinder

 2*pi*r = 6

      r = $\frac{6}{2pi}$

      r = $\frac{3}{pi}$ Inch

and the height = 10inch.

Volume of the resultant cylinder will be

   = $pi*r^2*h$

   = $pi*(\frac{3}{pi} )^2*10$

   = $\frac{9}{pi}*10$

   = $\frac{90}{pi}$ = 28.662 $in^3$

CASE-2: When the length of the rectangle is to used as the circumference of the resultant cylinder and the width of the cylinder is used as the height of the cylinder.

2*pi*r= 10

         r =  $\frac{5}{pi}$

and the height of the cylinder is 6 inch

Now the volume of the cylinder will be

 Volume= $pi*r^2*h$

              = $pi*(\frac{5}{pi} )^2*6$

              = $\frac{25}{pi}*6$

              = $\frac{150}{pi}$ = 47.77$in^3$

Clearly we have the second case in which the resulting Cylinder will have the maximum volume.

Therefore the CASE-2 will provide the maximum Volume of 47.77$in^3$ to the resulting cylinder in which the length of the rectangle is to be considered as the circumference of the Cylinder and the width of the rectangle is to be considered as the height of the cylinder.