Question 3, Exercise 5.3

Solutions of Question 3 of Exercise 5.3 of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.

A rectangular solid has a volume of 144 cubic units. The width is twice the height and the length is 2 units more than the width. Find the dimensions of the solid. (:!:Correction)

Solution.

Consider height = $x$ units
width = $2x$ units
length = $2x+2$ units
Volume = 144 cubic units.

By given condition $$\begin{align*} & x(2x)(2x+2) = 144 \\ \implies & 4x^2(x+1)=144 \\ \implies & x^2(x+1)=36 \\ \implies & x^3+x^2-36=0 \end{align*}$$

Suppose $$p(x)=x^3+x^2-36.$$ Since $$\begin{align*} p(3)&=3^3+3^2-36 \\ &=27+9-36 = 0 \end{align*}$$ This gives $x=3$ is zeros of $p(x)$. Thus

width = $2(3)$ = 6 units
length = $2(3)+2$ = 8 units
height = 3 units

Hence the dimension is 6 units by 8 units by 3 units.

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