Question

A hollow square shaped tube open at both ends is made of iron. The side of the internal square is \[5\;cm\] and the length of the tube is $ 8\;cm $ . There is $ 192\;c{m^3} $ of iron in this tube. Find its thickness.

Answer 1

Hint: Firstly check the side length of the external square and then the volume of the external square and equate it to the given volume of the tube and find the thickness of the tube.

Complete step-by-step answer:
A hollow square shaped tube open at both ends is made of iron. The side of the internal square is \[5\;cm\] and the length of the tube is $ 8\;cm $ . There is $ 192\;c{m^3} $ of iron in this tube.

Let us assume that the thickness of the tube is equal to $ x\;cm $ .
As per given the side of the internal square is \[5\;cm\]. So, the side of the external square is equal to $ \left( {5 + 2x} \right)\;cm $ .
As given that the length of the tube is equal to $ 8\;cm $ . The formula for the volume of the given tube is equal to the product of side square and length of tube.
So, the volume of the square is equal to
$ \left( {5 + 2x} \right)\left( {5 + 2x} \right)8\;c{m^3} $ .
The volume of the internal square is equal to $ 5 \times 5 \times 8 = 200\;c{m^3} $ .
Now as per given there is $ 192\;c{m^3} $ of iron in this tube. So, the difference of both the volumes is equal to $ 192\;c{m^3} $.
 $
  \left( {5 + 2x} \right)\left( {5 + 2x} \right)8 - 200 = 192 \\
  8{\left( {5 + 2x} \right)^2} = 392 \\
  {\left( {5 + 2x} \right)^2} = 49 \\
  \left( {5 + 2x} \right) = 7 \;
  $
Further simplify,
 $
  5 + 2x = 7 \\
  2x = 7 - 5 \\
  x = 1 \;
  $
Hence, the thickness of the tube is equal to $ 1\;cm $ .
So, the correct answer is “$ 1\;cm $”.

Note: Please make the diagram of the tube to avoid any confusion for the side length of the tube and of internal and external squares. Also use the formula for the volume of the cuboid as the product of length to breadth to height.