Question

The volume of cuboid is given by the product of it’s length, breadth and height. The length of cuboid is \[{\text{2}}{{\text{x}}^{\text{2}}}\] times its breadth and the height is \[\dfrac{{\text{3}}}{{\text{2}}}{\text{xy}}\] times of length. Find the volume of the cuboid if its breadth is \[{\text{6}}{{\text{y}}^{\text{2}}}\].

Answer 1

Hint: In order to find volume of cuboid we required the product of all of its three sides which are length, breadth and height. Let the breadth be x so length of cuboid is \[{\text{2}}{{\text{x}}^{\text{2}}}\] times its breadth and the height is \[\dfrac{{\text{3}}}{{\text{2}}}{\text{xy}}\] times of length. Using this useful information we can proceed to calculate the area of cuboid.

Complete step by step answer:

Diagram:

As we know the size of breadth which is given as \[{\text{6}}{{\text{y}}^{\text{2}}}\] and length of cuboid is \[{\text{2}}{{\text{x}}^{\text{2}}}\] times its breadth
So length of cuboid is \[{\text{(2}}{{\text{x}}^2}{\text{)(6}}{{\text{y}}^{\text{2}}}) = 12{{\text{x}}^2}{{\text{y}}^{\text{2}}}\]
And height is \[\dfrac{{\text{3}}}{{\text{2}}}{\text{xy}}\] times of length
So , height of cuboid is \[{\text{(}}\dfrac{{\text{3}}}{{\text{2}}}{\text{xy)12}}{{\text{x}}^{\text{2}}}{{\text{y}}^{\text{2}}}{\text{ = 18}}{{\text{x}}^{\text{3}}}{{\text{y}}^{\text{3}}}\]
Thus, as we know all the three values length , breadth and height if the cuboid we can easily calculate its volume by using \[{\text{v = l}}{\text{.b}}{\text{.h}}\]
\[
  \therefore {\text{v = (12}}{{\text{x}}^{\text{2}}}{{\text{y}}^{\text{2}}}{\text{)(6}}{{\text{y}}^{\text{2}}}{\text{)(18}}{{\text{x}}^{\text{3}}}{{\text{y}}^{\text{3}}}{\text{)}} \\
  {\text{ = 1296}}{{\text{x}}^{\text{5}}}{{\text{y}}^{\text{7}}} \\
 \]
Hence, the volume of cuboid \[{\text{ = 1296}}{{\text{x}}^{\text{5}}}{{\text{y}}^{\text{7}}}\].

Note: From the given data, one should properly form the equations and should not get confused between the length breadth and height. A cuboid is a 3D shape. Cuboids have six faces, which form a convex polyhedron. Broadly, the faces of the cuboid can be any quadrilateral. More narrowly, rectangular cuboids are made from 6 rectangles, which are placed at right angles. A cuboid that uses all square faces is a cube.