Two cups A and B are similar, cup A

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Hi, I have this question which is pretty easy but I forgot how do do it. Here is the question:
Two cups A and B are similar, cup A has a height of 30CM and cup B has a height of 20CM. Cup A has a volume of 54CM3. Calculate the volume of cup B.
 
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hypnagogue

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Assuming the cups are cylinders...

v = πr^2*h
54 = πr^2*30
r = √(54/30*π) = .757 cm

I assume by similar you mean the ratio of the cups' heights to their radiuses are the same? If so,

hA/rA = hB/rB
rB = hB*rA/hA = 20*.757/30 = .505

so

vB = π*(.505)^2*20 = 16.02 (approximately)
 
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Thanks but is there a simpler method of explaining it?
 

HallsofIvy

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As long as the cups have the same "shape" (are similar) then doubling, tripling, etc. any length does the same to the others.

The volume of any shape is arrived at by multiplying three lengths together (possibly times some constants- like (4/3)pi). Since doubling any length will double all three lengths in the calculation, the volume will be multiplied by 2*2*2= 8. In other words: the volume is multiplied by the cube of the length multiplier. (That's why volume is given in cm3 when length is in cm.)

In your problem A has height 30 cm and B has height 20 cm: to go from A to B, multiply the height by 2/3. Since the two cups are "similar", all lengths are multiplied by 2/3 and so the volume is multiplied by (2/3)^3= 8/27.

Since the volume of A is 54 cm3, the volume of B is
(8/27)(54) cm3.

I get exactly 16 cm3.
 

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