A rectangular sheet of perimeter 27 cm and dimensions x cm by y cm is to be rolled into a cylinder. What values of x and y give the largest volume?

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aruv | High School Teacher | (Level 2) Valedictorian

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A rectangular sheet of perimeter 27 cm and dimensions x cm by y cm is to be rolled into a cylinder. What values of x and y give the largest volume?

A rectangular sheet is `x cm xx ycm.`

So

Perimeter of rectangular sheet= 2 (x+y)     (i)

But given perimeter of rectangle=27           (ii)

Therefore from (i) and (ii), we have

2(x+y)=27

x+y=13.5                      (iii)

Let sheet is rolled  about breadth of the rectangular sheet to form cylinder. So height of the cylinder will be y cm.

Thus

The circumference of the base of cylinder = x cm

Let radius of the base of the cylinder be r, so

`2pir=x`

`r=x/(2pi)`

Thus, the volume V of the cylinder will be

`V=pir^2h`

`=pi(x/(2pi))^2y`

`=1/(4pi)x^2y`  

Substitute y from (iii), we have

`V=1/(4pi)x^2(13.5-x)`

Differentiate V with respect to x,

`(dV)/(dx)=(1/(4pi))(2x(13.5-x)-x^2)`

`=(1/(4pi))(27x-3x^2)`

For maximum or minimum, 

`(dV)/(dx)=0`

`=>27x-3x^2=0`

`=>3x(9-x)=0`

`` Either  3x=0 or `9-x=0`

`=> x=0 ,9`

Since x is length of the rectangular sheet , so x=0 not possible.

`=> x=9`

To test the maxima, apply second derivative test. So we have

`(d^2V)/(dx^2)=(1/(4pi))(27-6x)`

`((d^2V)/(dx^2))_(x=9)=(1/(4pi))(27-54)=(-27)/(4pi)<0`

Thus when x=9, cylinder has maximum volume.

So x=9 and y=4.5 give the largest volume of the cylinder. 

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aruv's profile pic

aruv | High School Teacher | (Level 2) Valedictorian

Posted on

A rectangular sheet of perimeter 27 cm and dimensions x cm by y cm is to be rolled into a cylinder. What values of x and y give the largest volume?

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