Question

How many possible dimensions (in natural numbers) of a rectangle with a perimeter 36 cm
pls tell me fast it will rlly help​

Answer 1

Answer:

First of all, find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible, when revolved about one of its sides. We will also find the maximum volume.

Step-by-step explanation:

Let x,y be the length and breadth of the rectangle

Given,

2x+2y=36

x+y=18

⇒y=18−x

Let the rectangle is being resolved about its length y and Volume of the cylinder is given by,

V=πx2y

=πx2(18−x)

∴V=π(18x2−x3)

differentiating the above equation, we get,

dxdV​=π(36x−3x2)

equate dxdV​=0

⇒π(36x−3x2)=0

solving the above equation, we get,

∴x=12

y=18−12

∴y=6

again differentiating the above equation, we get,

dx2d2V​=π(36−6x)

dx2d2V​→x=12=π(36−6(12))=−36π<0 maxima

For x=12 volume of cylinder is maximum.

So, the dimensions of rectangle are x=12cm,y=6cm

Maximum volume of resultant cylinder;

V=π[18(12)2−(12)3]

∴V=864πcm

or V=2712.96