



 
A "mathematical" formula of course! :) Thomas, the formula for the volume of a cone is 1/3 pi R^{2} 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 k^{2} H^{3}. If we calculate the same thing for half the volume, we multiply by 1/2 on both sides and we get this: Now, we can think of the smaller cup as having volume v, and a new height (h) and a new mouth radius (r). But we know it is the same shape as the large cup, so r = kh here as well: 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: Divide out the extra terms: 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: A. According to our calculation, it should be about 6.37 / 1.26 = 5.06 units. V = 1/3 pi R^{2} H v = 1/3 pi r^{2} h So it works. If you want half the volume with the same shape of conical cup, divide the height by 1.26. Cheers,  


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