Find the volume of a mathematically similar glass when given the height and volume.

Question

There are two mathematically similar glasses. One has a height of 16cm and a volume of 375cm³. The other has a height of y cm and a volume of 192cm³. Find the height, y.

I thought here because they are mathematically similar you would just put them in a ratio and solve for y (like I did in a similar question with correct results). So 16:375 = y:192 then rearrange and solve for y. I didn’t get the right answer and any assistance would be GREATLY appreciated.

Answer 1

Volume = V = hA = hpi^2 = 375= 16A

Area of the cylinder's bottom = 375/16 = 23.4375 sq. cm.

I think they want to shrink the height in the same proportion as they shrink the radius or Area of the bottom of the cylinder.

If they kept the bottom area and radius the same, then you'd be right, and all the shrinkage would have to be in the height, so h or y = about 8.19

But if you make the ratio of height to bottom area the same, then you get 11.51 for the height of the smaller cylinder shaped glass. Or if you keep the ratio of height to radius of the glass bottom, then about 12.85 for the height of the smaller glass.

Take an easier problem: say a square is 2 by 4 with Area = 8. Then you want another square with Area = 4, half as much. the sides won't be half as much, 1 by 2. But if you kept the 2 sides in the same ratio, 2:4, 1:2 then the sides of the new smaller square would be s:2s where s(2s) = 4 or s^2 = 2 and s=1.414 with other side=2s=2.828

So, do something similar with 3 dimensions

Let the height to bottom Area remain the same. For a volume of 372, the ratio is 16/372/16 = about .69

then the smaller glass would have a ratio of height to bottom of .69 = y/192/y = y^2/192 or y^2 = 192(.69) = about 132, y=sqr132 = 11.5

Do the same with a ratio of height to radius of the glass bottom and it seems to come out closer to 12.85

"mathematically similar" suggests a ratio of y to bottom area or y to bottom radius. Either way, y is in the range of 11.5 to 12.85. This all assumes the glass is shaped like a cylinder, like a can.