Height and Major Radius of a Frustum Given Volume, Minor Radius, and draft Angle

[Asgoranth]

As the title states, I am working on a project that requires a given volume of sand to fit within a cone-shaped 'hopper' that is essentially an inverted Frustum (truncated cone) I know the following:
- The bottom diameter of the hopper should be 300mm (r)
- The Hopper should hold 3.36L when full (V)
- The walls of the hopper should be angled at 60deg (theta)

I am aware that V= ((pi*H)/3)*(R^2 + r^2 + Rr) but trying to rearrange this to set both H and R in terms of r has not really gotten me anywhere.

It seems intuitively true that given Volume (v), minor radius (r), and draft angle (theta) one should be able to calculate height (H) and Major Radius (R) but for the life of me I cant figure it out.

I have been able to figure that volume of the 'cut off' portion of the cone is ~ 2.04L=pi*150^2*(86.6/3), which makes the volume for a complete cone 2.04L+3.36L = 5.4L However, for this 'complete' cone all I have to work with is total volume and draft angle, which seems to be less than what I started with. I have tried searching for a formula that finds the height and radius of a cone given its angle and volume, but so far no luck.

Any help would be greatly appreciated!

 

[Deleted member 4993]

As the title states, I am working on a project that requires a given volume of sand to fit within a cone-shaped 'hopper' that is essentially an inverted Frustum (truncated cone) I know the following:
- The bottom diameter of the hopper should be 300mm (r)
- The Hopper should hold 3.36L when full (V)
- The walls of the hopper should be angled at 60deg (theta)

I am aware that V= ((pi*H)/3)*(R^2 + r^2 + Rr) but trying to rearrange this to set both H and R in terms of r has not really gotten me anywhere.

It seems intuitively true that given Volume (v), minor radius (r), and draft angle (theta) one should be able to calculate height (H) and Major Radius (R) but for the life of me I cant figure it out.

I have been able to figure that volume of the 'cut off' portion of the cone is ~ 2.04L=pi*150^2*(86.6/3), which makes the volume for a complete cone 2.04L+3.36L = 5.4L However, for this 'complete' cone all I have to work with is total volume and draft angle, which seems to be less than what I started with. I have tried searching for a formula that finds the height and radius of a cone given its angle and volume, but so far no luck.

Any help would be greatly appreciated!

Can you draw a sketch of the frustum labelling the bottom diameter of the hopper, the angle of the walls of the hopper ?

 

[Dr.Peterson]

As the title states, I am working on a project that requires a given volume of sand to fit within a cone-shaped 'hopper' that is essentially an inverted Frustum (truncated cone) I know the following:
- The bottom diameter of the hopper should be 300mm (r)
- The Hopper should hold 3.36L when full (V)
- The walls of the hopper should be angled at 60deg (theta)

I am aware that V= ((pi*H)/3)*(R^2 + r^2 + Rr) but trying to rearrange this to set both H and R in terms of r has not really gotten me anywhere.

It seems intuitively true that given Volume (v), minor radius (r), and draft angle (theta) one should be able to calculate height (H) and Major Radius (R) but for the life of me I cant figure it out.

I have been able to figure that volume of the 'cut off' portion of the cone is ~ 2.04L=pi*150^2*(86.6/3), which makes the volume for a complete cone 2.04L+3.36L = 5.4L However, for this 'complete' cone all I have to work with is total volume and draft angle, which seems to be less than what I started with. I have tried searching for a formula that finds the height and radius of a cone given its angle and volume, but so far no luck.

I'm not sure why you aren't using the volume formula you stated. You don't need to work with the "complete cone", though that does seem on the surface like a good idea, considering the data you have.

Express the top radius R as a function of H, using the known angle, and then substitute that into either volume formula. You'll then have a cubic equation in H, which you will probably want to use some technology to solve for H.

Also, don't forget to convert the volume from liters to cubic centimeters (or millimeters, if you prefer).

 

[Asgoranth]

Hello Mr. Kahn!

I'm seeking equation for R and h as circled
r = 15cm
h1 = 15sqrt(3)cm = 25.98cm
V1 = pi*(15^2)*(25.98/3)=6121.39cc
V2 = 3355.71cc
V1+V2 = 9477.10cc
Tip angle = 30deg

 

[Asgoranth]

I'm not sure why you aren't using the volume formula you stated. You don't need to work with the "complete cone", though that does seem on the surface like a good idea, considering the data you have.

Express the top radius R as a function of H, using the known angle, and then substitute that into either volume formula. You'll then have a cubic equation in H, which you will probably want to use some technology to solve for H.

Also, don't forget to convert the volume from liters to cubic centimeters (or millimeters, if you prefer).

I gave your suggestion a shot, but I may have not done it properly. substituting (h/sqrt(3)) for R into the equation I wrote above ultimately gives:

Which doesn't seem to have any realistic solutions for v2 =3355.71cc

Would it be possible to treat f(x) =x*sqrt(3) as a solid of revolution?

 

[Dr.Peterson]

substituting (h/sqrt(3)) for R into the equation I wrote above

I'm not sure which equation you are referring to. You've mentioned too many different things.

If h here refers to the height of the frustum (not the complete cone), R isn't 0 when h is 0. Rather, it's 15+h/sqrt(3) cm.

Please show more details of your work.

Would it be possible to treat f(x) =x*sqrt(3) as a solid of revolution?

If you need to derive the formula for volume of a cone or of a frustum. But you already have those, don't you?

 

[Asgoranth]

Figured it out! Its easy, just a few steps.

basically just

1) found all the parameters for the "missing" cone using given small radius, and cone angle. This was easier because the angle is 30deg.

2) Used the total volume of the cone (frustum + "missing" cone) to derive the total height.

3) Used the total Height to find Big R of Frustum

4) Used total height of cone minus "missing" cone height to determine height of Frustum

5) using Frustum height, R and r, was able to check to ensure that volume returned as given

Thanks for the replies, for some reason I was determined to make this problem harder than it needed to be! Hopefully this can help someone in the future!