A square piece of tin of side 24 cm is to be made into a box without top by cutting a square from
each corner and folding up the flaps to forms a box.
(i) The length, breadth and height of the box formed, in terms of x are
(a) (24 – 2x, x, 24 – 2x) (b) (24 – 2x, 24 – 2x, x)
(c) (x, 24 – 2x, 24 – 2x) (d) (x, x, x,)
(ii) Volume V of the box expressed in terms of x is
(a) (24 – 2x)3
(b) x3
(c) x2
(24 – 2x) (d) x(24 – 2x)2
(iii) Maximum value of volume of box is
(a) 256 cm3
(b) 512 cm3
(c) 1024 cm3
(d) 2048 cm3
(iv) The value of x when volume is maximum is
(a) 12 cm (b) 8 cm (c) 4 cm (d) 2 cm
(v) The cost of box, if rate of making the box is Rs. 5 per cm2
, when volume is maximum is
(a) Rs. 2840 (b) Rs. 3840 (c) Rs. 2040 (d) Rs. 3480
Answers
Given: A square piece of tin of side 24 cm is to be made into a box without top by cutting a square of side x from each corner and folding up the flaps to forms a box.
To Find : The length, breadth and height of the box formed, in terms of x are
Solution:
Length = Breadth = 24 - 2x
Height = x
Hence The length, breadth and height of the box formed, in terms of x are
are (b) (24 – 2x, 24 – 2x, x)
Volume V of the box expressed in terms of x is
length * breadth * height
= (24 - 2x)²x
V = (24 - 2x)²x
dV/dx = (24 - 2x)² + x 2(24 - 2x) (-2)
= (24 - 2x) (24 - 2x - 4x)
= (24 - 2x)(24 - 6x)
dV/dx = 0
=> x= 12 and x = 4
x = 12 is not possible as then length = breadth = 0
so x = 4
d²V/dx² = (24 - 2x)(-6 ) + ( - 2 )(24 - 6x)
x = 4 => -96 < 0 hence maximum volume at x = 4
V = (24 - 2x)²x
x = 4
=> V = 16² (4) = 1024 cm³
value of x when volume is maximum is 4 cm
16 , 16 , 4 are the dimensions
area = 16 * 16 + 2 * 4 * (16 + 16)
= 256 + 256
= 512 cm²
cost of box = 512 * 5 = 2560 Rs
none of the given option matches
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