A cylinder and a cone have the same volume. The cylinder has radius x and height y. The cone has radius 3x. Find the height of the cone in terms of y.
Answers
The volume of cylinder = πr²h
Given that, the radius of the cylinder is x and height is y.
Therefore,
Volume of cylinder = π(x)²y = πx²y
Now,
The volume of cone = 1/3 πr²h
Given that, the radius of the cone is 3x.
So, the volume of cone = 1/3 π(3x)²h
“A cylinder and a cone have the same volume.”
According to question,
⇒ πx²y = 1/3 π(3x)²h
π cancel out throughout, we left with
⇒ x²y = 1/3 × 9x² × h
⇒ x²y = 3x² × h
⇒ y = (3x² × h)/x²
⇒ y = 3h
⇒ h = y/3
Therefore, the height of the cone in terms of y is y/3.
Given :-
- Volume of cylinder = Volume cone .
- Radius of Cylinder = x
- height of cylinder = y
- Radius of cone = 3x .
Formula used :-
- Volume of Cylinder = π * r² * h
- volume of cone = (1/3) * π * r² * h
Solution :-
Let height of cone is H .
Putting all values and comparing Both Volume we get,
→ π * r² * h = (1/3) * π * r² * h
→ π * x² * y = (1/3) * π * (3x)² * H
Cancel π from both sides ,
→ x² * y = (1/3) * 9x² * H
→ x² * y = 3x² * H
Cancel x² from both sides ,
→ y = 3H
Dividing both sides by 3 ,