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A bottle, full of olive oil (light grey) and balsamic 11 Aug 2006, 11:24
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A bottle, full of olive oil (light grey) and balsamic vinegar (dark grey), is made from two right cylinders whose radii are in the ratio 5:2. Currently, when the bottle is standing on its larger base, the depth of the oil is x cm. If the bottle is inverted and the contents allowed to settle , the depth of the oil will rise to y cm. What is the height of the bottle in centimeters terms of x and y?



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A bottle, full of olive oil (light grey) and balsamic 11 Aug 2006, 11:43
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H = (4/9) (25x/4 - y)

If this is the answer, I can explain.
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A bottle, full of olive oil (light grey) and balsamic 11 Aug 2006, 12:18
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Explain how you got the answer- whether it's right or wrong, it may help others!
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A bottle, full of olive oil (light grey) and balsamic 11 Aug 2006, 12:44
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Phew!!! Its time consuming but not difficult. If I encounter such question in real thing, I will answer it in at max 30 seconds and move on.

To solve equate the volumes of vinegar and olive oil when bottle is upright and when upside down.

If I have not made the calculation mistakes then

Height = (54x+25y)/29
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A bottle, full of olive oil (light grey) and balsamic 11 Aug 2006, 13:03
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let:

r1 = radius of smaller cylinder
r2 = radius of larger cylinder
h1 = height of smaller cylinder
h2 = height of larger cylinder

x = height of oil from one large cylinder base
y = height of oil after you turn it upside down

V1 = Volume from larger bottom
V2 = Volume from smaller bottom

Given:

r2/r1 = 5/2

h1+h2 = H

Find Volume from both sides

V1 = pi*h1*r1^2 + pi*(x-h1)*r2^2

V2 = pi*h2*r2^2 + pi*(y-h2)*r1^2

since V1 = V2

and r2/r1 = 5/2

with some manipulation u get:

H = (4/9)(25x/4 - y)
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A bottle, full of olive oil (light grey) and balsamic 11 Aug 2006, 22:58
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The easiest thing to do is focus on the red wine vinegar!
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A bottle, full of olive oil (light grey) and balsamic 12 Aug 2006, 13:24
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Seems right... but I'm still struggling to understand it.
Isn't is as simple as r1^2*h1=r2^2*x?



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