Difference between revisions of "2013 AIME I Problems/Problem 7"

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==Solution==
 
==Solution==
<math>\boxed{041}</math>
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After using the pythagorean formula three times, we can quickly see that the sides of the triangle are 10, <math>\sqrt{(x/2)^2 + 64}</math>, and <math>\sqrt{(x/2)^2 + 36}</math>. Therefore, we can use Heron's formula to set up an equation for the area of the triangle.
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The semi perimeter is (10 + <math>\sqrt{(x/2)^2 + 64}</math> + <math>\sqrt{(x/2)^2 + 36}</math>)/2
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900 = <math>\frac{1}{2}</math>((10 + <math>\sqrt{(x/2)^2 + 64}</math> + <math>\sqrt{(x/2)^2 + 36}</math>)/2)((10 + <math>\sqrt{(x/2)^2 + 64}</math> + <math>\sqrt{(x/2)^2 + 36}</math>)/2 - 10)((10 + <math>\sqrt{(x/2)^2 + 64}</math> + <math>\sqrt{(x/2)^2 + 36}</math>)/2 - <math>\sqrt{(x/2)^2 + 64}</math>)((10 + <math>\sqrt{(x/2)^2 + 64}</math> + <math>\sqrt{(x/2)^2 + 36}</math>)/2 - <math>\sqrt{(x/2)^2 + 36}</math>).
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Solving, we get <math>\boxed{041}</math>.
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=2013|n=I|num-b=6|num-a=8}}
 
{{AIME box|year=2013|n=I|num-b=6|num-a=8}}

Revision as of 16:52, 18 March 2013

Problem 7

A rectangular box has width $12$ inches, length $16$ inches, and height $\frac{m}{n}$ inches, where $m$ and $n$ are relatively prime positive integers. Three faces of the box meet at a corner of the box. The center points of those three faces are the vertices of a triangle with an area of $30$ square inches. Find $m+n$.

Solution

After using the pythagorean formula three times, we can quickly see that the sides of the triangle are 10, $\sqrt{(x/2)^2 + 64}$, and $\sqrt{(x/2)^2 + 36}$. Therefore, we can use Heron's formula to set up an equation for the area of the triangle.

The semi perimeter is (10 + $\sqrt{(x/2)^2 + 64}$ + $\sqrt{(x/2)^2 + 36}$)/2

900 = $\frac{1}{2}$((10 + $\sqrt{(x/2)^2 + 64}$ + $\sqrt{(x/2)^2 + 36}$)/2)((10 + $\sqrt{(x/2)^2 + 64}$ + $\sqrt{(x/2)^2 + 36}$)/2 - 10)((10 + $\sqrt{(x/2)^2 + 64}$ + $\sqrt{(x/2)^2 + 36}$)/2 - $\sqrt{(x/2)^2 + 64}$)((10 + $\sqrt{(x/2)^2 + 64}$ + $\sqrt{(x/2)^2 + 36}$)/2 - $\sqrt{(x/2)^2 + 36}$).

Solving, we get $\boxed{041}$.

See also

2013 AIME I (ProblemsAnswer KeyResources)
Preceded by
Problem 6
Followed by
Problem 8
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All AIME Problems and Solutions
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