What are the possible expressions for the dimensions of the cuboids whose volumes are given below?<br/><br/>(i) Volume: $3 x^{2}-12 x$ <br/><br/>(ii) Volume: $12 k y^{2}+8 k y-20 k$

Solution:

Volume of cuboid $=$ Length $\times$ Breadth $\times$ Height

The expression given for the volume of the cuboid has to be factorised. One of its factors will be its length, one will be its breadth, and one will be its height.

(i) $2 x^{2}-12 x=3 x(x-4)$

One of the possible solutions is as follows.

Length $=3$, Breadth $=x$, Height $=x-4$

(ii) $12 k y^{2}+8 k y-20 k=4 k\left(3 y^{2}+2 y-5\right)$

$=4 k\left[3 y^{2}+5 y-3 y-5\right]$

$=4 k[y(3 y+5)-1(3 y+5)]$

$=4 k(3 y+5)(y-1)$

One of the possible solutions is as follows.

Length $=4 k$, Breadth $=3 y+5$, Height $=y-1$