What are the possible expressions for the dimensions of the cuboid whose volumes are given below?
(i) Volume: ${3 x}^{2} - 12 x$

(ii) Volume: $12 k y^{2} + 8 k y - 20 k$

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Solution

(i)Volume: ${3 x}^{2} - 12 x$

We know that Volume of Cuboid = length × breadth × height

Volume of cuboid = ${3 x}^{2} - 12 x$

We need to factorize this given polynomial to find values of length, breadth and height.

${3 x}^{2} - 12 x$

=3x(x−4)

Therefore, one possibility is that length =3, breadth =x and height =(x−4)

(ii)Volume: $12 k y^{2} + 8 k y - 20 k$

We know that Volume of Cuboid = length × breadth × height

Volume of cuboid = $12 k y^{2} + 8 k y - 20 k$

We need to factorize this given polynomial to find values of length, breadth and height.

$12 k y^{2} + 8 k y - 20 k$

= $12 k y^{2} + 20 k y - 12 k y - 20 k$

=4ky(3y+5)−4k(3y+5)

=(3y+5)(4ky−4k)

=(3y+5)4k(y−1)=4k(y−1)(3y+5)

Therefore, one possibility is that length =4k, breadth =(y−1) and height =(3y+5)

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12 k y^{2} + 8 k y - 20 k

\begin{matrix} \begin{matrix} V o l u m e : 3 x^{2} - 12 x \end{matrix} (i) \end{matrix} \begin{matrix} \begin{matrix} V o l u m e : 12 k y^{2} + 8 k y - 20 k \end{matrix} (i i) \end{matrix}

V o l u m e : 12 k y^{2} + 8 k y - 20 k

What are the possible expressions for the dimensions of cuboid , whose volume is : 12ky^2 +8ky- 20 k?

12 k x^{2} + 8 k x - 20 k ?

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