Question

if c,t and v are curved surface area total surface area and volume of a cylinder then show that th2= ch2 +4v2+8v2rh

Answer 1

Given :

For A cylinder

The curved surface area = c

The total surface area = t

The volume = v

To Proved :

t h² = c h² + 4 v² + 8 v² r h

Solution :

We know that

Curved surface area of cylinder = 2 π r h

    where r = radius

               h = height

So,   c =  2 π r h      .........1

And

Total  surface area of cylinder = 2 π r h + 2 π r²

    where r = radius

               h = height

So,     t = 2 π r h + 2 π r²         ,............2

And

The volume of cylinder = π r² h

     where r = radius

               h = height

So,     v =  π r² h                 ........3

Now,

The dimensions of both t h² and c h² are $L^{4} M^{0} T^{0}$

And The dimensions of 4 v² is $L^{6} M^{0} T^{0}$

And  The dimensions of 8 v²r h is $L^{8} M^{0} T^{0}$

So, we have to prove t h² =  c h² + 2 v h

Therefor

From Right Hand Side of equation

i.e c h² + 2 v h = ( 2 π r h ) × h² + 2 ×  ( π r² h ) ×h

                          = 2 π r h × h² + 2 π r² h²

                          =  2 π r h³ +  2 π r²  h²

From Left Hand Side of equation

i.e  t h²  = (  2 π r h + 2 π r² ) × h²

             =    2 π r h × h² +  2 π r²  × h²

             = 2 π r h³ +  2 π r²  h²

Hence,   c h² + 2 v h =  t h²   Proved    Answer