What are the possible expressions for the dimensions of the cuboids whose volumes are given below? Volume : $12 k y^{2} + 8 k y - 20 k$

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Solution

Step 1. Factorize the given expression of volume.
$V = 12 k y^{2} + 8 k y - 20 k \Rightarrow V = 4 k [3 y^{2} + 2 y - 5] (∵ 4 k is common) \Rightarrow V = 4 k [3 y^{2} + 5 y - 3 y - 5] \Rightarrow V = 4 k [y (3 y + 5) - 1 (3 y + 5)] \Rightarrow V = 4 k (3 y + 5) (y - 1)$
Step 2. Find the possible dimension of the cuboid.
The volume of the cuboid is computed by using the formula,
$V = length \times breadth \times height$
Compare with the volume computed in step 1.
the dimensions of the volume are
$4 k, 3 y + 5 and (y - 1)$
Hence, the possible dimensions of the cuboid are,
$4 k, 3 y + 5 and (y - 1)$

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

\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}

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