A box maker wants to design an open box from a piece of cardboard measuring 24 cm x 24 xm by cutting off
equal squares from each corner's and turning up the sides as shown below.
X
24 cm
24 cm
Based on above information answer the following:
Let the length of side of each square cut off from the corner be x cm then height of open box is
(a) (24 – x) cm (b) (12 – x) cm
(c) x cm
(d) 2x cm
Volume of the open box so formed is
(a) 4x? – 96x + 576
(c) 4X? – 96x + 576
(b) 4x? - 96x2 + 576x
(d) None of these
The base area of open box so formed is
(a) 4X? + 576 – 96x
(b) 4x2 = 576 + 96x
(c) 4x² + 96x – 576
(d) 4x2 - 96x – 576
(iv)
For maximum volume of the box, i should be
(a) 2 cm
(b) 3 cm
(c) 4 cm
(0) 6 cm
(v)
The maximum volume of box is
(a) 256 cm?
(b) 256 cm
(c) 1024 crn?
(d) 1024 cm
Answers
Answer:
1.option (c)- x cm, 2. option (b) 4x³-96x²+576x, 3. option (a) 4x²-96x+576, 4. option (c) 4cm, 5. option(c) 1024cm³.
Step-by-step explanation:
Given : A cardboard measuring 24cm ×24cm and squares of equal measures are cut from its corner and turned up the sides to form an open box.
length of the side of each square cut off from the corner is x cm.
To find :
height of the open box so formed.
Volume of the open box .
The base area of open box so formed .
For maximum volume of the box, x should be?
The maximum volume of box.
Solution:
The open box so formed is a cuboid of length = (24-2x)cm, breadth = (24-2x)cm and height = x cm.
1. Now the height of the open box option (c) x cm.
2. volume of open box = volume of cuboid.
= l × b × h
= (24-2x)×(24-2x)× x
=[24×24-(2x×24)-(24×2x)+4x²]× x
=[576 - 48x - 48x + 4x²]× x
=[576 - 96x + 4x²]× x
=4x³ - 96x²+576x.
so , volume of open box so formed is option (b) 4x³ - 96x² + 576x.
3.Now the base area of the open box so formed is a square again with length = (24-2x) cm
base area of the open box so formed = side × side
=(24-2x) × (24-2x)
=576-48x-48x+4x²
=576 - 96x + 4x².
so, the base area of open box so formed is option (a) 4x²-96x+576.
4.For maximum value of the box, x should be option(c) 4cm.
To find the maximum volume of box , we have to check by putting the values for x, given in the option, in the volume .
5.By putting x =4,we get the the maximum volume of box .
volume of open box = 4x³-96x²+576x
= 4×(4³) - 96(4²) + 576×4
= 4×64 - 96×16 + 2304
=256 - 1536 + 2304
=1024cm³.
For x=4, the volume becomes maximum. The other three options doesn't give maximum volume.
∴The maximum volume of box is option(c) 1024cm³.
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