Difference between revisions of "2000 AIME I Problems/Problem 8"

Revision as of 23:25, 24 March 2011

Problem

A container in the shape of a right circular cone is $12$ inches tall and its base has a $5$ -inch radius. The liquid that is sealed inside is $9$ inches deep when the cone is held with its point down and its base horizontal. When the liquid is held with its point up and its base horizontal, the height of the liquid is $m - n\sqrt [3]{p},$ from the base where $m,$ $n,$ and $p$ are positive integers and $p$ is not divisible by the cube of any prime number. Find $m + n + p$ .

Solution

Solution 1

The scale factor is uniform in all dimensions, so the volume of the liquid is $\left(\frac{3}{4}\right)^{3}$ of the container. The remaining section of the volume is $\frac{1-\left(\frac{3}{4}\right)^{3}}{1}$ of the volume, and therefore $\frac{\left(1-\left(\frac{3}{4}\right)^{3}\right)^{1/3}}{1}$ of the height when the vertex is at the top.

So, the liquid occupies $\frac{1-\left(1-\left(\frac{3}{4}\right)^{3}\right)^{1/3}}{1}$ of the height, or $12-12\left(1-\left(\frac{3}{4}\right)^{3}\right)^{1/3}=12-3\left(37^{1/3}\right)$ . Thus $m+n+p=\boxed{052}$ .

Solution 2

(Computational) The volume of a cone can be found by $V = \frac{\pi}{3}r^2h$ . In the second container, if we let $h',r'$ represent the height, radius (respectively) of the air (so $12 -h'$ is the height of the liquid), then the volume of the liquid can be found by $\frac{\pi}{3}r^2h - \frac{\pi}{3}(r')^2h'$ .

By similar triangles, we find that the dimensions of the liquid in the first cone to the entire cone is $\frac{3}{4}$ , and that $r' = \frac{rh'}{h}$ ; equating,

$\begin{align*}\frac{\pi}{3}\left(\frac{3}{4}r\right)^2 \left(\frac{3}{4}h\right) &= \frac{\pi}{3}\left(r^2h - \left(\frac{rh'}{h}\right)^2h'\right)\\ \frac{37}{64}r^2h &= \frac{r^2}{h^2}(h')^3 \\ h' &= \sqrt[3]{\frac{37}{64} \cdot 12^3} = 3\sqrt[3]{37}\end{align*}$

Thus the answer is $12 - h' = 12-3\sqrt[3]{37}$ , and $m+n+p=\boxed{052}$ .

@@ Line 19: / Line 19: @@
 h' &= \sqrt[3]{\frac{37}{64} \cdot 12^3} = 3\sqrt[3]{37}\end{align*}</cmath>
-Thus the answer is <math>12 - h' = 12-3\sqrt[3]{37}</math>, and <math>m+n+p=52</math>.
+Thus the answer is <math>12 - h' = 12-3\sqrt[3]{37}</math>, and <math>m+n+p=\boxed{052}</math>.
 == See also ==
 {{AIME box|year=2000|n=I|num-b=7|num-a=9}}
 [[Category:Intermediate Geometry Problems]]

2000 AIME I (Problems • Answer Key • Resources)
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