A drainage tile is a cylindrical shell 21 cm long. The inside and outside diameters are 4.5 cm and 5.1 cm, respectively. What is the volume of clay required for making a tile?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(30.24\pi\) cm3
The clay forms a hollow cylinder, so its volume is the volume of the outer cylinder minus that of the hollow inside: \(V=\pi R^2h-\pi r^2h=\pi h(R^2-r^2)\), where \(R\) and \(r\) are the outer and inner radii.
The radii and the length (each radius is half its diameter):
Outer radius \(R=5.1\div2=2.55\,\mathrm{cm}\)
Inner radius \(r=4.5\div2=2.25\,\mathrm{cm}\)
Length \(h=21\,\mathrm{cm}\)
The volume of clay (by the difference of two squares, the sum times the difference of the radii):
\(R^2-r^2=(R+r)(R-r)=4.8\times0.3=1.44\)
Putting \(h\) and \(R^2-r^2\) in the formula: \(V=\pi\times21\times1.44=30.24\pi\,\mathrm{cm^3}\)