Given a right circular cone with the line of symmetry along x = 0 , and the base along

Jay Barrett

Jay Barrett

Answered question

2022-05-08

Given a right circular cone with the line of symmetry along x = 0, and the base along y = 0, how can I find the maximum volume paraboloid (parabola revolved around the y-axis) inscribed within the cone? Maximising the volume of the paraboloid relative to the volume of the right circular cone. In 2-D, the parabola has 2 points of tangency to the triangle, one of each side of the line of symmetry. I have tried using the disk method to find the volume of the cone, and the parabola, both with arbitrary equations such as y = b a x, and y = c d x 2 , but I end up with a massive equation for several variables, instead of a simple percentage answer. Any help is appreciated! Thanks in advance.

Answer & Explanation

Jaylon Richmond

Jaylon Richmond

Beginner2022-05-09Added 10 answers

Hint:

Note that here a and b are given.

The tangency condition is
a 2 4 d ( b c ) = 0
If you decide to work with c , then
d = a 2 4 ( b c )
The volume of the paraboloids, inscribed in the cone, is
V ( c ) = π 0 c c y d d y = π 0 c ( c y ) 4 ( b c ) a 2 d y = 2 π a 2 ( b c 2 c 3 )
Then solve
V ( c ) = 0
to find the largest volume for
c = 2 3 b
Verify that
V m a x = 8 9 V c o n e

Do you have a similar question?

Recalculate according to your conditions!

New Questions in High school geometry

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?