Question 1136735
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The volume of the cylinder is<br>
{{{(pi)(r^2)(h) = (pi)(4^2)(12) = 192(pi)}}}<br>
The portion of the cone that is inside the cylinder is similar to the whole cone.  The radii of those two cones are in the ratio 2:3; since the cones are similar, the ratio of their heights is also 2:3.  That makes the height of the small cone 2/3 of 9, which is 6.  Then the volume of the cone that is inside the cylinder is<br>
{{{(1/3)(pi)(r^2)(h) = (1/3)(pi)(4^2)(6) = 32(pi)}}}<br>
ANSWER 1: The volume of water that spills from the cylinder when the cone is lowered into it is the volume of the small cone: 32pi (cm^3).<br>
The volume of water that spilled from the {{{highlight(cross(cone))}}} cylinder, 32pi, is 1/6 of the volume of the {{{highlight(cross(cone))}}} cylinder.  So when the cone is removed, the volume of water in the cylinder will be 5/6 of the original volume.  Then since the cylinder has constant radius, the height of the water in the cylinder after the cone is removed is 5/6 of the full height of the cylinder.<br>
ANSWER 2: The height of water in the cylinder after the cone is removed is (5/6)*12 = 10cm.<br>
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Thanks to tutor @Ikleyn for noticing the errors in my original response, now fixed<br>