Question

what are the possible expressions for the dimensions of the cuboids whose volumes are (b) 12 ky^2 + 18 ky - 20k​

Answer 1

Solution:-

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.

Thus, the possible expressions for the dimensions of the cuboids whose volume are 4k, (y - 1), (3y + 5).

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