If the height of two right circular cylinders are in the ratio 3 : 4 and the perimeters are in the ratio 1 : 2 , then find tthe ratio of their volumes.
Answers
Step-by-step explanation:
Given :-
The height of two right circular cylinders are in the ratio 3 : 4 and the perimeters are in the ratio 1 : 2 .
To find :-
Find tthe ratio of their volumes?
Solution :-
Given that
The ratio of heights of the two circular cylinders = 3:4
Let they be 3X units and 4X units
Let the height of the first cylinder
= 3X units
Let the height of the second cylinder
= 4X units
We know that
Perimeter of a right circular cylinder
= 2(πd+h) units
Where , d = diameter , h = height
The ratio of the perimeters
=> 2(πd+3X):2(πd+4X)
On cancelling 2 then
=> (πd+3X):(πd+4X)
According to the given problem
The ratio of the perimeters of the cylinders = 1:2
=> (πd+3X):(πd+4X) = 1:2
=> (πd+3X)/(πd+4X) = 1/2
=> 2(πd+3X) = 1(πd+4X)
=> 2πd+6X = πd+4X
=> 2πd-πd = 4X-6X
=> πd = -2X
=> d = -2π/X
=> 2r = -2π/X
=> r = -π/X
Therefore, radius of the cylinder
= -π/X units
Volume of the cylinder = πr²h cubic units
Volume of the first cylinder
=> π(-π/X)²(3X)
=> π(π²/X²) (3X)
=> 3Xπ³/X²
=> 3π³/X Cubic Units
Volume of the second cylinder
= πr²h cubic units
=> π(-π/X)²(4X)
=> π(π²/X²)(4X)
=>4X π³/X²
=> 4π³/X Cubic Units
Their ratio of the volumes of the two cylinders
= 3π³/X : 4π³/X
=> (3π³/X) /( 4π³/X)
=> 3/4
=> 3:4
Answer:-
The ratio of the volumes of the two cylinders is 3:4
Used formulae:-
- Perimeter of a right circular cylinder
= 2(πd+h) units
Where , d = diameter , h = height
- Volume of the cylinder = πr²h cubic units