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the radio of two cylinder are in the ratio of 1: √3 if the volume of two cylinder be same , find the ratio of their respective height?
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Understanding the Problem
To find the ratio of the heights of two cylinders with equal volumes and given radii in the ratio of 1:√3, we start by recalling the formula for the volume of a cylinder.
Volume of a Cylinder
The volume (V) of a cylinder is given by the formula:
V = πr²h
where r is the radius and h is the height of the cylinder.
Given Conditions
- Let the radius of the first cylinder be r1 = 1.
- Let the radius of the second cylinder be r2 = √3.
- Both cylinders have the same volume, V.
Setting Up the Equations
Using the volume formula for both cylinders:
1. For the first cylinder:
- V1 = π(1)²h1 = πh1
2. For the second cylinder:
- V2 = π(√3)²h2 = 3πh2
Since the volumes are equal (V1 = V2), we set the equations equal to each other:
πh1 = 3πh2
Simplifying the Equation
By cancelling out π from both sides, we get:
h1 = 3h2
Finding the Ratio of Heights
To find the ratio of the heights, we can express h1 in terms of h2:
- h1/h2 = 3/1
Thus, the ratio of the heights of the two cylinders is:
Final Ratio
- The ratio of the heights of the two cylinders is 3:1.
This means the height of the first cylinder is three times that of the second cylinder.
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?the radio of two cylinder are in the ratio of 1: √3 if the volume of two cylinder be same , find the ratio of their respective height?
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