Question

18. For the box to satisfy certain requirements, its length must be three metre
greater than the width, and its height must be two metre less than the width.
(i) If width is taken as x, which of the following polynomial represent
volume of box ?

Page 5 of 9

(a) x
3
-5x-6 (b) x3
+x2
-6x

(c) x
3
- 6x2
-6x (d) x3
+x-6

(ii) Which of the following polynomial represent the area of paper sheet used
to make box ?
(a) x2
-5x-6 (b) 6x
2
+4x-12

(c) x
3
-6x2
-6x (d) 6x2
+3x-4

(iii)If it must have a volume of 18 unit, what must be its length?
(a) 6 unit (b) 3 unit
(c) 4 unit (d) 2 unit
(iv) At a volume of 18 cubic unit, what must be its height?
(a) 1 unit (b) 3 unit
(c) 2 unit (d) 4 unit
(v) If box is made of a paper sheet which cost rupees100 per square unit,
what is the cost of the paper?
(a) Rs 5400 (b) Rs 10800
(c) Rs 2700 (d) Rs 3400

Answer 1

Given:

Let width of box(b) = x

Then length(l) = x+3

And Height(h) = x-2

Solution:

(i) Answer:  (b) $x^{3} +x^{2} -6x$    

We know that volume of a rectangular box = length×width×height

                                                                       = $x(x+3)(x-2)$

                                                                       = $(x^{2} +3x)(x-2)$

                                                                       = $x^{2} (x-2)+3x(x-2)$

                                                                       = $x^{3} -2x+3x^{2} -6x$

                                    Volume of the box    = $x^{3} +x^{2} -6x$

(ii) Answer:  (b) $6x^{2} +4x-12$

The area of paper sheet required to make the box = Total surface area of box = 2(lh+bh+lb)

⇒$2[(x+3)(x-2)+x(x-2)+(x+3)x]$

⇒$2[x^{2} +x-6+x^{2} -2x+x^{2} +3x]$

⇒$2[3x^{2} +2x-6]$

⇒$6x^{2} +4x-12$

∴The area of paper used = $6x^{2} +4x-12$

(iii) Answer: (a) 6 units

Given that, volume = 18 units

We have to find its length

We have already calculated volume =  $x^{3} +x^{2} -6x$ = 18

⇒ $x^{3} +x^{2} -6x-18=0$

Solving above equation we get x= 3, √2i-2, -√2i-2

So x= width = 3 units

Then length = x+3 = 3+3 = 6 units  

(iv) Answer: (a) 1 unit

Given volume = 18 cubic units

We got width = 3 units when volume = 18

so height = x-2 units = 3-2 1 unit

Height = 1 unit

(v) Answer: (a) Rs.5400

Cost of paper = Rs.100 per sq.unit

We already found that Area of paper required =   $6x^{2} +4x-12$

when x= 3, Area = 54 sq.units

Total cost = 54 × 100 = Rs.5400

Answer 2

Answer:

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