A solid iron cuboid \(4.4\,\text{m}\times2.6\,\text{m}\times1\,\text{m}\) is recast into a hollow cylindrical pipe of internal radius 30 cm and thickness 5 cm. Find the length of the pipe.
112 m
Step 1: Volume of the cuboid
The cuboid is given with dimensions: length = 4.4 m, breadth = 2.6 m, height = 1 m.
Volume of cuboid = length × breadth × height
= \(4.4 \times 2.6 \times 1 = 11.44 \, \text{m}^3\)
Step 2: Understand the pipe structure
The pipe is hollow. This means it has an outer cylinder and an inner cylinder.
Step 3: Cross-sectional area of the hollow pipe
Cross-sectional area of hollow cylinder (annulus) = \(\pi (R^2 - r^2)\)
= \(\pi (0.35^2 - 0.30^2)\)
= \(\pi (0.1225 - 0.0900)\)
= \(\pi (0.0325)\)
= \(0.0325\pi \, \text{m}^2\)
Step 4: Volume of the pipe
Volume of hollow cylinder = cross-sectional area × length
= \(0.0325\pi \times L\)
Step 5: Equating volumes
The cuboid is recast into the pipe, so volumes are equal:
\(11.44 = 0.0325\pi L\)
Step 6: Solve for length
\(L = \dfrac{11.44}{0.0325\pi}\)
\(L \approx 112\, \text{m}\)
Final Answer: The length of the pipe is 112 m.