Answer:
about 5.83 cm
Explanation:
When water filling a cylinder of diameter 8 cm and height 12 cm is dumped into a cone of height 20 cm and radius 9 cm, you want to know how far below the top of the cone the water will reach.
Volumes
The volume of the cylinder is ...
V = πr²h
V = π(4 cm)²(12 cm) = 192π cm³
The volume of the cone is ...
V = 1/3πr²h
V = 1/3π(9 cm)²(20 cm) = 540π cm³
Similar shapes
The water filling part of the cone will have a shape similar to that of the whole cone. The volume of the cone is ...
(540π)/(192π) = 45/16 = 2.8125
times that of the cylinder. Hence the filled volume of the cone will be ...
1/2.8125
times that of the whole cone.
For similar shapes, the ratio of volumes is the cube of the ratio of their linear dimensions. That means the filled height of the cone will be ...
∛(1/2.8125) ≈ 1/1.411554
times the full height of the cone. In cm, that is ...
(20 cm)/1.411554 ≈ 14.1688 cm
Space
The height of the empty space above the filled portion of the cone is then ...
20 cm -14.1688 cm = 5.8312 cm
The cone will be fill to within about 5.83 cm of the top.
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Additional comment
An alternative way to get to the same answer is to write the volume of the cone in terms of its height:
V = 1/3π(9/20h)²h = 27π/400h³
Setting this volume equal to the cylinder volume, we can solve for h:
27πh³/400 = 192π
h = ∛(192·400)/3 = (8/3)∛150 ≈ 14.17 . . . . cm
As above, the space height is the difference between this height and 20 cm.
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