Question 1, Exercise 5.3

Solutions of Question 1 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.

The volume of a drinking water bottle is 120 cubic centimeters. The bottle is 7 centimeters longer than it is width and it is 3 cm wider than it length. Find the dimensions of the bottle. ( Correction)

Solution.

Consider length of bottle = $x$ cm
Width of bottle = $x+3$ cm
Height of bottle = $x+3+7=x+10$ cm
Volume of bottle = $120 cm^3$

By given condtion, we have $$\begin{align*} & x(x+3)(x+10)=120 \\ \implies & x(x^2+3x+10x+30)-120=0\\ \implies & x^3+13x^2+30x-120=0. \end{align*}$$

Consider $$p(x)=x^3+13x^2+30x-120$$ Now $$\begin{align*} p(2)&=2^3+13(2)^2+30(2)-120 \\ &=8+52+60-120 =0 \end{align*}$$ This gives $x=2$ is zero of $p(x)$. Hence

Length of bottle = $2 cm$
Width of bottle = $2+3=5cm$
Height of bottle = $2+10=12 cm$

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