Question

Obtain the volume of rectangular boxes with the following length, breadth and height respectively.
 $ \left( i \right)5a,3{a^2},7{a^4} $
 $ \left( {ii} \right)2p,4q,8r $
 $ \left( {iii} \right)xy,2{x^2}y,2x{y^2} $
 $ \left( {iv} \right)a,2b,3c $

Answer 1

Hint: Rectangular boxes are in the shape of a cuboid. Use the volume of a cuboid formula to obtain the volume of the rectangular boxes.
Formula used: Volume of a cuboid is $ l \times b \times h $ where ‘l’ is the length of the cuboid, ‘b’ is the breadth of the cuboid and ‘h’ is the height of the cuboid.

Complete step-by-step answer:
 $ \left( i \right)5a,3{a^2},7{a^4} $
In the first case we are given that the length, breadth, height of a cuboid are $ 5a,3{a^2},7{a^4} $ respectively.
Volume of the cuboid is $ l \times b \times h $ where $ l = 5a,b = 3{a^2},h = 7{a^4} $
Volume of the cuboid
 $
   = 5a \times 3{a^2} \times 7{a^4} \\
   = 5 \times {a^1} \times 3 \times {a^2} \times 7 \times {a^4} \\
   = 5 \times 3 \times 7 \times {a^1} \times {a^2} \times {a^4} \\
   = 105 \times {a^{1 + 2 + 4}} \\
  \left( {\because {a^m} \times {a^n} = {a^{m + n}}} \right) \\
   = 105 \times {a^7} \\
   = 105{a^7} \\
  $
Therefore, the volume of the cuboid with the length, breadth, height $ 5a,3{a^2},7{a^4} $ respectively is $ 105{a^7} $ cube units.
 $ \left( {ii} \right)2p,4q,8r $
In the second case we are given that the length, breadth, height of a cuboid are $ 2p,4q,8r $ respectively.
Volume of the cuboid is $ l \times b \times h $ where $ l = 2p,b = 4q,h = 8r $
Volume of the cuboid
 $
   = 2p \times 4q \times 8r \\
   = 2 \times p \times 4 \times q \times 8 \times r \\
   = 2 \times 4 \times 8 \times p \times q \times r \\
   = 64 \times pqr \\
   = 64pqr \\
  $
Therefore, the volume of the cuboid with the length, breadth, height $ 2p,4q,8r $ respectively is $ 64pqr $ cube units.
 $ \left( {iii} \right)xy,2{x^2}y,2x{y^2} $
In the third case we are given that the length, breadth, height of a cuboid are $ xy,2{x^2}y,2x{y^2} $ respectively.
Volume of the cuboid is $ l \times b \times h $ where $ l = xy,b = 2{x^2}y,h = 2x{y^2} $
Volume of the cuboid
 $
   = xy \times 2{x^2}y \times 2x{y^2} \\
   = x \times y \times 2 \times {x^2} \times y \times 2 \times x \times {y^2} \\
   = 2 \times 2 \times {x^1} \times {x^2} \times {x^1} \times {y^1} \times {y^1} \times {y^2} \\
   = 4 \times {x^{1 + 2 + 1}} \times {y^{1 + 1 + 2}} \\
  \left( {\because {a^m} \times {a^n} = {a^{m + n}}} \right) \\
   = 4 \times {x^4} \times {y^4} \\
   = 4{x^4}{y^4} \\
  $
Therefore, the volume of the cuboid with the length, breadth, height $ xy,2{x^2}y,2x{y^2} $ respectively is $ 4{x^4}{y^4} $ cube units.
 $ \left( {iv} \right)a,2b,3c $
In the fourth case we are given that the length, breadth, height of a cuboid are $ a,2b,3c $ respectively.
Volume of the cuboid is $ l \times b \times h $ where $ l = a,b = 2b,h = 3c $
Volume of the cuboid
 $
   = a \times 2b \times 3c \\
   = a \times 2 \times b \times 3 \times c \\
   = 2 \times 3 \times a \times b \times c \\
   = 6 \times abc \\
   = 6abc \\
  $
Therefore, the volume of the cuboid with the length, breadth, height $ a,2b,3c $ respectively is $ 6abc $ cube units.

Note: A cube is also a type of cuboid where the length, breadth and height of a cube are equal. l=b=h.
A cuboid is a polyhedron having six faces, eight vertices and twelve edges. The faces of the cuboid are parallel. But all the faces of a cuboid may not be equal in dimensions.