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The volume of a cylinder is 48.125 cm3, which is formed by rolling a rectangular paper sheet along the length of the paper. If cuboidal box (without any lid i.e., open at the top) is made from the same sheet of paper by cutting out the square of side 0.5 cm from each of the four corners of the paper sheet, then what is the volume of this box

let the width of the sheet of paper be x
let its length be y
so the cylinder made from it would have a height of y
the width would be its circumference
2πr = x
r = x/(2π)
volume = πr^2 h = π(x^2/(4π^2) y = x^2 y/(4π)
48.125 = x^2 y/(4π)
192.5π = x^2y
y = 192.5/x^2
now to the box:
length = (y1)
width = (x1)
height = .5
volume = .5(x1)(y1) = .5(x1)(192.5/x^2  1)
= (1/2)(192.5/x  x  192.5/x^2 + 1)
getting nowhere .....
not enough information
in the first part, there is no unique solution for x and y
which would change the shape of the sheet of paper, which would change the volume of the resulting box
posted by Reiny

We do have a way for this question.Only way is to think out of the box. Let me tell you how it goes:
Volume of cylinder= (pi) (r)(r)(h)= 48.125
(r)(r)h = 15.3125
15.3125 is not a perfect square. To make it perfect square, h should be 5cm.
Now (r)(r)=15.3125/5 = 3.0625 which makes r = 1.75cm
Therefore, rectangular sheet has length of 2(pi)(r)=11cm and width (h) of 5cm.
After cutting a square of 0.5cm from each corner, the dimension of the cuboid becomes:
Length=111=10cm; Breadth=51=4cm; Height=0.5cm.
Therefore, Volume of cuboid= 10 x 4 x 0.5 = 20 cubic cm.
Thank you for reading it!
Have a nice day Reiny!
Jay Bankotiposted by Jay Bankoti
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