Question 1143219
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The volume of the cup is<br>
{{{V = (1/3)(pi)(r^2)(h) = (1/3)(pi)(4^2)(12) = 64(pi)}}}<br>
a. If the cup is filled to a depth of 8cm, that is 2/3 the full depth, so the radius of the water in the cup will be 2/3 of the full radius, which is 8/3cm.  The volume of water is then<br>
{{{V = (1/3)(pi)((8/3)^2)(8) = (512/27)pi}}}<br>
Since the question asks for an answer as the nearest whole number, perform the calculation and round as required.<br>
Note that, though the problem gives a hint about using similar triangles, there is a much faster way to answer this question.  If the height of the water in the cup is 2/3 the full height of the cup, then the volume of water in the cup is (2/3)^3 of the total volume:<br>
{{{((2/3)^3)(64pi) = (512/27)pi}}}<br>
b. This question is answered far more easily using the concept noted above.<br>
If the volume is 1/2 of the total volume, then the height of the cone (depth of the water) is the full height, multiplied by the CUBE ROOT of 1/2.<br>
{{{h = 12*(1/2)^(1/3) = 9.5244}}} to 4 decimal places.<br>
Than round as directed.