A rectangular sheet of paper 30 cm X 18cm can be transformed into a right circular cylinder in two ways - by rolling the paper along its length or by rolling it along its breadth . Find the ratio of the volume of two cylinders formed.
rolling along length:
height = 30, circumference = 18
2πr = 18
r = 9/π
volume = π(81/π^2)(30) = 2430/π
rolling along breadth
height = 18
circumf = 30
2πr = 30
r = 15/π
Volume = π(225/π^2)(18) = 4050/π
ratio of volume breadth : volume of height
=(4050/π) : 2430/π
= 5 : 3
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According to the question
First case
L:30 cm
B:18 cm
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To find the ratio of the volumes of the two cylinders formed, we first need to calculate the volumes of each cylinder.
Let's consider rolling the paper along its length first:
When rolled along its length, the length of the paper becomes the height (h) of the cylinder, and the breadth of the paper becomes the diameter (d) of the base of the cylinder.
Given that the length of the paper is 30 cm and the breadth is 18 cm, we have:
Length of the cylinder (l) = 30 cm
Breadth of the cylinder (b) = 18 cm
The height (h) of the cylinder is equal to the length of the paper, so h = 30 cm.
The radius (r) of the base of the cylinder is half the breadth of the paper, so r = 18/2 = 9 cm.
Using the formulas for the volume of a cylinder, we have:
Volume of the cylinder rolled along the length = π * r^2 * h
Volume of the first cylinder = π * 9^2 * 30
Next, let's consider rolling the paper along its breadth:
When rolled along its breadth, the breadth of the paper becomes the height (h) of the cylinder, and the length of the paper becomes the diameter (d) of the base of the cylinder.
Given that the length of the paper is 30 cm and the breadth is 18 cm, we have:
Length of the cylinder (l) = 18 cm
Breadth of the cylinder (b) = 30 cm
The height (h) of the cylinder is equal to the breadth of the paper, so h = 18 cm.
The radius (r) of the base of the cylinder is half the length of the paper, so r = 30/2 = 15 cm.
Using the formulas for the volume of a cylinder, we have:
Volume of the cylinder rolled along the breadth = π * r^2 * h
Volume of the second cylinder = π * 15^2 * 18
Now, we can find the ratio of the volumes of the two cylinders:
Ratio = Volume of the first cylinder / Volume of the second cylinder
Ratio = (π * 9^2 * 30) / (π * 15^2 * 18)
Simplifying this expression:
Ratio = (9^2 * 30) / (15^2 * 18)
Ratio = (81 * 30) / (225 * 18)
Ratio = 2430 / 4050
Therefore, the ratio of the volume of the two cylinders formed is 2430/4050 = 2/3.