Question

1. what are the possible expressions for the dimensions of the cuboids whose volumes are given below?

(i) volume : $3x^2 - 12x$
(ii) volume : $12ky^2 + 8ky - 20k$

Explanation:

Part (i)

Step1: Factor out the GCF

The volume is \(3x^2 - 12x\). The greatest common factor (GCF) of \(3x^2\) and \(12x\) is \(3x\).
\(3x^2 - 12x = 3x(x - 4)\)

Step2: Determine dimensions

For a cuboid, volume \(V=\text{length}\times\text{width}\times\text{height}\). Since we have factored the volume into \(3x\times(x - 4)\times1\) (we can consider 1 as one of the dimensions as \(3x(x - 4)\times1=3x(x - 4)\)).

Part (ii)

Step1: Factor out the GCF

The volume is \(12ky^2 + 8ky - 20k\). The GCF of \(12ky^2\), \(8ky\) and \(-20k\) is \(4k\).
\(12ky^2 + 8ky - 20k=4k(3y^2 + 2y - 5)\)

Step2: Factor the quadratic

Factor \(3y^2 + 2y - 5\). We need two numbers \(a\) and \(b\) such that \(a\times b=3\times(-5)=-15\) and \(a + b = 2\). The numbers are \(5\) and \(-3\).
\(3y^2+2y - 5=3y^2+5y-3y - 5=y(3y + 5)-1(3y + 5)=(3y + 5)(y - 1)\)

Step3: Determine dimensions

Now, the factored form of the volume is \(4k\times(3y + 5)\times(y - 1)\). So the dimensions can be \(4k\), \(3y + 5\) and \(y - 1\).

Answer:

(i) The possible dimensions are \(3x\), \(x - 4\) and \(1\) (or any permutation of these, e.g., \(x - 4\), \(3x\), \(1\) etc.).
(ii) The possible dimensions are \(4k\), \(3y + 5\) and \(y - 1\) (or any permutation of these, e.g., \(3y + 5\), \(4k\), \(y - 1\) etc.).