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What are the possible expressions for the dimensions of the cuboids whose volume are given below?
i) Volume: 3x² - 12x ii) Volume: 12ky² + 8ky - 20k

Solution:

i) Volume of Cuboid = 3x² - 12x

But we know that, Volume of Cuboid = length × breadth × height

Hence, we shall express the given polynomial as the product of three expressions using factorization.

3x² - 12x = 3x(x - 4)

Thus, the possible expressions for the length, breadth and height are:

Length = 3, breadth = x, height = x - 4

Length = 3, breadth = x - 4, height = x

Length = x, breadth = 3, height = x - 4

Length = x, breadth = x - 4, height = 3

Length = x - 4, breadth = x, height = 3

Length = x - 4, breadth = 3, height = x

ii) Volume of Cuboid = 12ky² + 8ky - 20k

But we know that, Volume of Cuboid = length × breadth × height

Hence, we shall express the given polynomial as the product of three expressions using factorization.

12ky² + 8ky - 20k = 4k(3y² + 2y - 5)

Now taking 3y² + 2y - 5 , find two numbers p, q such that:

p + q = co-efficient of y
pq = product of the co-efficient of y² and the constant term.

p + q = 2 (co-efficient of y)

pq = 3 × (-5) = -15 (product of the co-efficient of y² and the constant term.)

By trial and error method, we get p = 5, q = -3.

Now splitting the middle term of the given polynomial,

3y² + 2y - 5 = 3y² + 5y - 3y - 5

= 3y² - 3y + 5y - 5

= 3y( y -1) + 5( y - 1)

= (3y + 5) ( y - 1)

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

Thus, the possible expressions for the length, breadth and height is,

Length = 4k, breadth = y - 1, height = 3y + 5.

Length = 4k, breadth = 3y + 5, height = y - 1.

Length = y - 1, breadth = 4k, height = 3y + 5.

Length = y - 1, breadth = 3y + 5, height = 4k.

Length = 3y + 5, breadth = 4k, height = y -1.

Length = 3y + 5, breadth = y - 1, height = 4k.

☛ Check: NCERT Solutions Class 9 Maths Chapter 2

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