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Question
A rectangular metal sheet 36cm x 20cm can be formed into a right circular cylinder, either by rolling its length or by rolling along its breadth. Find the ratio of the volumes of the two cylinders thus formed.
Solution
Dimensions of rectangle = 36cm x 20cm
Let the rectangle be rolled along its length to form a cylinder, thus the length and breadth of the rectangle will be equal to circumference and height (h) of the cylinder respectively.
Let r be the radius of the cylinder.
Circumference of cylinder = 36cm
2 x π x r = 36
r = `(36)/(2π)`
r = `(18)/π"cm"`
thus,
Volume of the cylinder so formed
= π x r2 x h
= `π xx (18/π)^2 xx 20`
= `(6480)/π"cm"^3` ....................................(1)
Now,
Let the rectangular be rolled along its breadth to form a cylinder, thus the length and breadth of the rectangle will be equal to height (H) and circumference of the cylinder respectively.
Let R be the radius of the cylinder.
Circumference of cylinder = 20cm
2 x π x R = 20
R = `(20)/(2π)`
R = `(10)/π"cm"`
thus,
Volume of the cylinder so formed
= π x R2 x H
= `π xx (10/π)^2 xx 36`
= `(3600)/π"cm"^3` ...................................(2)
∴ Ratio of volumes of two cylinders
= `((1))/((2)`
= `(6480)/(3600)`
= 9 : 5.
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