Question

What length of a solid cylinder 2 cm in diameter must be taken to recast into a hollow cylinder of external diameter 20 cm, 0.25cm thick and 15 cm long?

Answer 1

Let The C1 is the smaller solid cylinder of height h₁.
radius = r₁ = diameter/2 = 2/2 = 1 cm.
now,
Volume of the C1 = V₁ = πr₁²h₁ = 3.14*(1)²h1  = 3.14.h₁
A/Q,
Solid Cylinder C₁ is recasted into a hollow Cylinder of height 15 cm.
diameter is 20 cm and its is 0.25 cm thick.
radius = r₂ = 20/2 = 10 cm.
height = h₂ = 15 cm.
thickness = 0.25 cm.
Volume of thick part o hollow Cylinder = π(r₂)²h₂ -  π(r₂ - 0.25)²h₂
                                                                  = πh₂{r₂² - (r₂ - 0.25)²}

                                                                  = 3.14*15 [ 25² - (25-0.25)² ]
                                                                  
                                                                  = 47.1[25² - (25² + 0.25²  - 2*25*0.25)]

                                                                  = 47.1( 25² -25² - 0.25² +2*25*0.25)

                                                                   = 47.1( 12.5 - 0.0625)
 
                                                                   = 47.1*12.4375

                                                                   =  585.80625

and,
The vol. of the small solid Cylinder C₁ = Vol. of the thick part of the Hollow cylinder.

⇒ 3.14h1 = 585.80625
h1 = 186.5625
Therefore the height of the small solid cylinder is 186.5625 cm.

Answer 2

thank you very much for your help