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I need to calculate the height of a glass(frustum) where the volume is half of total volume. Obviously, at h/2, volume will not be v/2. So my question is, at what height from the bottom of the glass is volume equal to half of full volume.

Where am I struck is the fact that 'R (bigger radius)' increases as the height.


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Let the radius of the top of the glass be $R$, and let the bottom radius be $r$, where $0<r<R$. Let the height of the glass be $h$. The glass can be extended to a cone. We will be working with several cones, all similar to each other. Since I do not remember the formula for the volume of a cone, we use scaling arguments.

Imagine a cone similar to our cones, but with top radius $1$. Choose units of volume so that this top radius $1$ cone has volume $1$. (That's OK, the unit of volume need not be simply related to the unit of length. There exists a country where distances are measured in feet but volumes in gallons.)

If we extend our glass to a complete cone, the volume of that complete cone is, by scaling, $R^3$. Similarly, the part of that cone which is beyond the glass has volume $r^3$. So the glass itself has volume $R^3-r^3$. Half of this is $(R^3-r^3)/2$. Add back the missing $r^3$ which lies beyond the glass. We conclude that the cone made up of the bottom half of the orange juice plus the stuff beyond the glass has volume $$\frac{R^3-r^3}{2}+r^3\quad\text{or more simply}\quad \frac{R^3+r^3}{2}.$$

It follows that when exactly half of the orange juice is removed from the glass, the top of the orange juice is a circle of radius $$\sqrt[3]{\frac{R^3+r^3}{2}}.$$ Pouring out juice until radius is just right sounds awkward. So let's work with heights. A similar triangle argument shows that the height of the "missing" cone is $$\frac{hr}{R-r}.$$ By scaling it follows that the height of the half-full of juice cone is $$\frac{h}{R-r}\sqrt[3]{\frac{R^3+r^3}{2}}.$$ To get the actual orange juice height when the glass is half-full, subtract the height of the missing cone. We get $$\frac{h}{R-r}\left(\sqrt[3]{\frac{R^3+r^3}{2}}-r\right).$$

Remark: Scaling arguments are somewhat underused in elementary mathematics. Physicists use them much more routinely.

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That can't be right! Putting $r=0$ in your final equation gives an answer of $0$, when the answer should be $h/\sqrt[3]2$. (And such glasses certainly exist.) – TonyK Apr 7 '12 at 19:07
@TonyK: Thanks, I didn't divide by $r$ when scaling. Fixed. The argument technically doesn't work when $r=0$. But the formula does. To make the argument work, one would scale down from $R$, but then there is an additional subtraction, since we then get the distance from the top of the glass, not the bottom. – André Nicolas Apr 7 '12 at 19:20
Now all you have to do is make it work when $r = R$. And such glasses certainly exist! – TonyK Apr 7 '12 at 19:35
Seriously, this formula is numerically unstable when $R$ and $r$ are nearly equal. But the answer is well-behaved. Do we need a separate (approximate) formula in this case, or is there a more stable formula? – TonyK Apr 7 '12 at 20:28
@TonyK: The limit as $r$ approaches $R$ is correct, kind of finding the volume of a cylinder the hard way. We can make things numerically stable by bringing the $R-r$ inside, and imitating, for numerical computation purposes, the proof that limit is right, – André Nicolas Apr 8 '12 at 0:30

A different approach. Your glass is a section of a cone, and the volume of a cone is $\frac \pi 3 r^2h$.

You also have a linear relationship $h=ar$ between the height and the radius (which comes fro the geometry of the situation).

The volume of your glass is the difference in volume between two cones, one with radius say $r_1$ (the smaller) and the other $r_2$ (larger). Putting this together the glass has volume

$$V =\frac {\pi a} 3 (r_2^3 - r_1^3)$$

You now need to find the volume for an arbitrary $r$ - which is of the same form as this, and find the value of $r$ which gives a volume $\frac V 2$. Then there is a little work to get the heights instead of the radii, and to measure the height from the base of the glass, not the apex of the cone.

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You can calculate the volume of any portion of your glass using integration.

Let $A(t)$ be the area of the section at height $t$. Let $y_0$ be the height of the orange juice in the glass.

You need to solve the following equation in $y$:

$$2\int_0^y A(t) dt = \int_0^{y_0} A(t) dt \,.$$

For the glass, the crosssection is a disk, so all you need is figure what is the radius as function in $t$ and integrate....

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This is no help! You know that the glass is a frustum, but you haven't used that fact at all. – TonyK Apr 7 '12 at 20:25

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