Question

How do you find the length, width, and height of a rectangular prism if the volume is $$h^3+h^2-20h$$ cubic meters?

Answer 1

Hint: A polynomial is factored completely when it is expressed as a product of one or more polynomials that cannot be factored further. To factor a polynomial completely, we need to identify the greatest common monomial factor. Not all polynomials can be factored in. We know that the volume of a rectangular prism is $$V=l.b.h$$.

Complete step by step answer:
As per the given question, we have to find the dimensions of the rectangular prism using factoring methods by factoring the given volume expression. Here, we have the given volume expression $$V=h^3+h^2-20h$$.

Let a rectangular prism with the dimension’s length ‘l’, width ‘b’ and height ‘h’ as shown in the figure below:

Then, the volume of the prism is given by $$V=l.b.h$$. But we know that the volume of the required prism is $$V=h^3+h^2-20h$$. Hence, we can combine both expressions. Then, we get
$$\Rightarrow l.b.h=h^3+h^2-20h$$.
Here, we can observe that ‘h’ is common on both sides of the equation. Thus, we can eliminate ‘h’ to get
$$\Rightarrow {\displaystyle \frac{l.b.h}{h}}={\displaystyle \frac{h^3+h^2-20h}{h}}\Rightarrow l.b=h^2+h-20$$
In the quadratic equation $$h^2+h-20$$, x-coefficient is 1. The product of $$x^2-\text{coefficient}$$ and the constant term is -20. We split up x-coefficient 1 into two numbers whose sum (or difference) is 1 and product is -20. Hence, the required numbers are 5 and -4. Thus, the equation becomes
$$\Rightarrow l.b=h^2+h-20=h^2+5h-4h-20$$
Taking $$(h+5)$$ common in the first 2 terms and last 2 terms, we get
$$\Rightarrow l.b=h(h+5)-4(h+5)=(h-4)(h+5)$$
As we know that length is greater than width, then we get $$l=(h+5)$$ meters and $$b=(h-4)$$ meters.
$$\therefore l=(h+5),b=(h-4)\text{ and }h=h$$ meters are the length, width and height of the rectangular prism respectively.

Note:
 In order to solve these types of questions, we need to have enough knowledge of factoring methods of polynomials. If polynomials can’t be factored then we can use quadratic formula $$x={\displaystyle \frac{-b\pm \sqrt{b^2-4ac}}{2a}}$$ to find the factors. We should avoid calculation mistakes to get the correct solution.