Volume of Concrete A culvert is constructed out of large cyl

Volume of Concrete A culvert is constructed out of large cylindrical shells cast in concrete, as shown in the figure. Using the formula for the volume of a cylinder given on the inside front cover of this book, explain why the volume of the cylindrical shell is V = piR^2/l - pir^2h Factor to show that V = 2pi average radius height thickness Use the \"unrolled\" diagram to explain why this makes sense geometrically.

Solution

Given V = piR^2h - pir^2h

Taing out pi and h as common factors

V = pi*( R^2 -r^2)h

Multiply and divide by 2:

V = 2pi*{(R^2 -r^2)/2}h

=2pi*{(R+r)(R-r)/2}h

= 2pi*(R+r)/2*(R -r)*h

= 2pi*Average radius*thickness*h

Avg radius = R+r/2 thickness= R-r height = h