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Q16(ii):
What are the possible expressions for the dimensions of the cuboids whose volumes are given below?
(ii) Volume : $12ky^2 + 8ky – 20k$
Solution :
Given Variables & Initial Setup
The volume of a cuboid is defined by the product of its three mutually perpendicular dimensions: length ($l$), width ($w$), and height ($h$).
Mathematically, this is expressed as:
$V = l \times w \times h$ [Per the geometric definition of a rectangular prism's volume]
We are given the volume of a cuboid as a polynomial expression in terms of variables $k$ and $y$:
$V(y) = 12ky^2 + 8ky - 20k$
To find the possible expressions for the dimensions, we must factorize this polynomial into three distinct linear factors, which will correspond to the length, width, and height.
Step 1: Extracting the Greatest Common Factor (GCF)
We begin by analyzing the terms of the polynomial: $12ky^2$, $8ky$, and $-20k$. We look for the highest common numerical coefficient and the highest degree of common variables.
- The numerical coefficients are $12$, $8$, and $-20$. The greatest common divisor (GCD) of these integers is $4$.
- Each term contains the variable $k$ to the first power.
- The variable $y$ is not present in the third term, so it cannot be factored out.
Factoring out the GCF ($4k$) from the entire expression yields:
$V(y) = 4k \left( \frac{12ky^2}{4k} + \frac{8ky}{4k} - \frac{20k}{4k} \right)$
$V(y) = 4k(3y^2 + 2y - 5)$
Step 2: Factorizing the Quadratic Expression
The expression inside the parentheses is a quadratic polynomial of the form $ay^2 + by + c$, where $a = 3$, $b = 2$, and $c = -5$. We will factorize this using the method of splitting the middle term.
We must find two numbers that satisfy two conditions simultaneously:
- Their product equals $a \times c = 3 \times (-5) = -15$.
- Their sum equals the middle coefficient $b = 2$.
The factors of $-15$ that add up to $2$ are $5$ and $-3$. We rewrite the middle term ($2y$) using these two numbers:
$3y^2 + 5y - 3y - 5$
Next, we group the terms into pairs to factor by grouping:
$(3y^2 - 3y) + (5y - 5)$
Factor out the common terms from each binomial group:
$3y(y - 1) + 5(y - 1)$
Since $(y - 1)$ is a common binomial factor, we can factor it out:
$(3y + 5)(y - 1)$
Step 3: Determining the Dimensions
Substituting the factorized quadratic back into our volume equation, we get the completely factorized form of the volume:
$V = 4k \cdot (3y + 5) \cdot (y - 1)$
Because the volume of a cuboid is the product of three dimensions ($l \times w \times h$), these three factors represent the possible expressions for the length, width, and height of the cuboid.
Geometric Visualization
Below is a precise oblique projection of the cuboid, illustrating how the three algebraic factors map to the spatial dimensions.
Final Solution: The possible expressions for the dimensions of the cuboid are $4k$, $(3y + 5)$, and $(y - 1)$.
More Questions from Class 9 Mathematics Polynomials EXERCISE 2.4
- Q1(i): Use suitable identities to find the following products: (i) $(x + 4) (x + 10)$
- Q1(ii): Use suitable identities to find the following products: (ii) $(x + 8) (x – 10)$
- Q1(iii): Use suitable identities to find the following products: (iii) $(3x + 4) (3x – 5)$
- Q1(iv): Use suitable identities to find the following products: (iv) $(y^2 + \frac{3}{2}) (y^2 – \frac{3}{2})$
- Q1(v): Use suitable identities to find the following products: (v) $(3 – 2x) (3 + 2x)$
- Q10(i): Factorise each of the following: (i) $27y^3 + 125z^3$ [Hint : See Question 9.]
- Q10(ii): Factorise each of the following: (ii) $64m^3 – 343n^3$ [Hint : See Question 9.]
- Q11: Factorise : $27x^3 + y^3 + z^3 – 9xyz$
- Q12: Verify that $x^3 + y^3 + z^3 – 3xyz = \frac{1}{2}(x + y + z)[(x – y)^2 + (y – z)^2 + (z – x)^2]$
- Q13: If $x + y + z = 0$, show that $x^3 + y^3 + z^3 = 3xyz$.
- Q14(i): Without actually calculating the cubes, find the value of each of the following: (i) $(–12)^3 + (7)^3 + (5)^3$
- Q14(ii): Without actually calculating the cubes, find the value of each of the following: (ii) $(28)^3 + (–15)^3 + (–13)^3$
- Q15(i): Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given: (i) Area : $25a^2 – 35a + 12$
- Q15(ii): Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given: (ii) Area : $35y^2 + 13y –12$
- Q16(i): What are the possible expressions for the dimensions of the cuboids whose volumes are given below? (i) Volume : $3x^2 – 12x$
- Q2(i): Evaluate the following products without multiplying directly: (i) $103 \times 107$
- Q2(ii): Evaluate the following products without multiplying directly: (ii) $95 \times 96$
- Q2(iii): Evaluate the following products without multiplying directly: (iii) $104 \times 96$
- Q3(i): Factorise the following using appropriate identities: (i) $9x^2 + 6xy + y^2$
- Q3(ii): Factorise the following using appropriate identities: (ii) $4y^2 – 4y + 1$
- Q3(iii): Factorise the following using appropriate identities: (iii) $x^2 – \frac{y^2}{100}$
- Q4(i): Expand each of the following, using suitable identities: (i) $(x + 2y + 4z)^2$
- Q4(ii): Expand each of the following, using suitable identities: (ii) $(2x – y + z)^2$
- Q4(iii): Expand each of the following, using suitable identities: (iii) $(–2x + 3y + 2z)^2$
- Q4(iv): Expand each of the following, using suitable identities: (iv) $(3a – 7b – c)^2$
- Q4(v): Expand each of the following, using suitable identities: (v) $(–2x + 5y – 3z)^2$
- Q4(vi): Expand each of the following, using suitable identities: (vi) $(\frac{1}{4}a - \frac{1}{2}b + 1)^2$
- Q5(i): Factorise: (i) $4x^2 + 9y^2 + 16z^2 + 12xy – 24yz – 16xz$
- Q5(ii): Factorise: (ii) $2x^2 + y^2 + 8z^2 – 2\sqrt{2}xy + 4\sqrt{2}yz – 8xz$
- Q6(i): Write the following cubes in expanded form: (i) $(2x + 1)^3$
- Q6(ii): Write the following cubes in expanded form: (ii) $(2a – 3b)^3$
- Q6(iii): Write the following cubes in expanded form: (iii) $(\frac{3}{2}x + 1)^3$
- Q6(iv): Write the following cubes in expanded form: (iv) $(x - \frac{2}{3}y)^3$
- Q7(i): Evaluate the following using suitable identities: (i) $(99)^3$
- Q7(ii): Evaluate the following using suitable identities: (ii) $(102)^3$
- Q7(iii): Evaluate the following using suitable identities: (iii) $(998)^3$
- Q8(i): Factorise each of the following: (i) $8a^3 + b^3 + 12a^2b + 6ab^2$
- Q8(ii): Factorise each of the following: (ii) $8a^3 – b^3 – 12a^2b + 6ab^2$
- Q8(iii): Factorise each of the following: (iii) $27 – 125a^3 – 135a + 225a^2$
- Q8(iv): Factorise each of the following: (iv) $64a^3 – 27b^3 – 144a^2b + 108ab^2$
- Q8(v): Factorise each of the following: (v) $27p^3 – \frac{1}{216} – \frac{9}{2}p^2 + \frac{1}{4}p$
- Q9(i): Verify : (i) $x^3 + y^3 = (x + y) (x^2 – xy + y^2)$
- Q9(ii): Verify : (ii) $x^3 – y^3 = (x – y) (x^2 + xy + y^2)$
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