Medicine Cup Frustum

Date: 04/25/2002 at 00:10:39
From: Andrew Clifford
Subject: Math - Frustum

I have a frustum (medicine cup); the smaller end has a diameter of 
2cm, the larger end has a diameter of 5cm, the height from A to C 
(| on diagram) is not given, and the angle (x65 on diagram) in the 
corner of the larger end is 65 degrees. I know that 1cm squared = 1ml 
and along the line AC i am required to give measurements 5ml, 10ml, 
15ml, 20ml, 25ml, and 30ml. 
                  
                      dia 5cm
                  _______A_______
                  \      |   x65/
                   \     |     /
                    \    |    /
                     \   |   /
                      \__|__/
                         C
                      dia 2cm

I am confused and have come to you as a last resort. Could you please 
help me out?

Thanks,
Andrew.

Date: 04/25/2002 at 12:23:19
From: Doctor Peterson
Subject: Re: Math - Frustrum

Hi, Andrew.

Here's something that might make this easier. Rather than using the 
formula for volume of a frustum (which you could find in our FAQ), 
since you have to work with a variable height frustum, you can work 
with the complete cone instead. Use the given angle to find the height 
h of the cone you would have to add to the bottom to reconstruct the 
cone of which this is a part:

      +------+------+ -------
       \     |     /       |
        \    |    /        |
         \   |   /         |H
          +--+--+ -------  |
           \ | /       |   |
            \|/        |h  |
             + --------------

You can use similar triangles to find H, the height of the whole cone. 
The volume of the whole frustum is the difference of the two cones.

Now suppose the cup is filled to a depth x. The you can use the cone 
of height h+x; again use similar triangles to find its base, and you 
can write an equation that gives the volume V for a given x. Solve 
this for x in terms of V, and you can answer the questions.

There are a lot of steps in this, and I don't know which will be hard 
for you, so write back and show me how far you get if you need more 
help.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/