the base radius of two cones are in the ratio 3 : 4 and their volumes are equal. find the ratio of their Heights?
Answers
Step-by-step explanation:
Given:-
The base radius of two cones are in the ratio 3 : 4
and their volumes are equal.
To find:-
Find the ratio of their Heights?
Solution:-
The ratio of the radius of the base of the two cones = 3:4
Let they be 3X and 4X units
Let the Radius of the first cone = 3X units
Let the radius of the second cone = 4X units
Let the height of the first cone be h1 units
And The height of the second cone be h2 units
Volume of a cone = (1/3)πr^2h cubic units
Volume of the first cone V1
=>(1/3)×π×(3X)^2×h1 cubic units
=>V1 = (1×π×9X^2×h1)/3 cubic units
V1 = 3πX^2h1 cubic units ------------------(1)
Volume of the second cone = V2
=>(1/3)×π×+4X)^2×h2 cubic units
=>V2 = (1×π×16X^2×h2)/3 cubic units
V2 =(16πX^2h2)/3 cubicunits---------------(2)
Given that
Volumes of the two cones are equal.
=>(1)=(2)
=>V1 = V2
=>3πX^2h1 = ( 16πX^2h2 ) /3
=>3(3πX^2 h1) = 16 πX^2 h2
=>9πX^2 h1) = 16 πX^2 h2
On cancelling πX^2 both sides then
=>9 h1 = 16 h2
=>9h1/h2 = 16
=>h1/h2 = 16/9
=>h1 : h2 = 16:9
Ratio of their Heights = 16:9
Answer:-
The ratio of the heights of the two cones = 16:9
Used formulae:-
- Volume of a cone = (1/3)πr^2h cubic units
Where 'r' is the radius and 'h' is the height and π=22/7