   SEARCH HOME Math Central Quandaries & Queries  Question from Thomas, a student: Im having trouble with a question. What kind of formula would i use to find out the length of a cup when its volume is halved? A "mathematical" formula of course! :)

Thomas, the formula for the volume of a cone is 1/3 pi R2 H, where h is the height of the cup and r is the radius at the mouth.

Since the radius varies directly as the height (in other words, the shapes are identical but the sizes of the cups are different), R = kH.

So V = 1/3 pi (kH)2 H = 1/3 pi k2 H3.

If we calculate the same thing for half the volume, we multiply by 1/2 on both sides and we get this:
1/2 V = 1/6 pi k2 H3.

Now, we can think of the smaller cup as having volume v, and a new height (h) and a new mouth radius (r).
v = 1/3 pi r2 h.

But we know it is the same shape as the large cup, so r = kh here as well:
v = 1/3 pi (kh)2 h = 1/3 pi k2 h3.

And the new cup is half the volume of the old cup, so v = 1/2 V. This means the two equations are equal to each other:
1/6 pi k2 H3 = 1/3 pi k2 h3

Divide out the extra terms:
1/2 H3 = h3

So if we want to know what h has to be in terms of H in order to make the volume half, we just take the cube root of both sides:

h = H / cuberoot(2)

The cube root of 2 is about 1.26. So we'd divide the initial height by 1.26 to get the smaller height.

Example:
Q. A large conical cup has a mouth radius of 3 units and a height of 6.37 units. If you cut off the top of the cup so it holds only half the volume, what is its new height?

A. According to our calculation, it should be about 6.37 / 1.26 = 5.06 units.
Let's check:
Calculate k:
R = kH
k = R/H
k = 3 / 6.37
k = 0.471

V = 1/3 pi R2 H
V = 1/3 (3.14) (3)2 (6.37)
V=~ 60 cubic units

v = 1/3 pi r2 h
v = 1/3 pi (kh)2 h
v = 1/3 (3.14) [(0.471)(5.06) ]2 (5.06)
v =~ 30 cubic inches

So it works. If you want half the volume with the same shape of conical cup, divide the height by 1.26.

Cheers,     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.