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Difference between revisions of "2005 AMC 12A Problems/Problem 22"

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<math>\mathrm{(A)}\ 8\qquad \mathrm{(B)}\ 10\qquad \mathrm{(C)}\ 12\qquad \mathrm{(D)}\ 14\qquad \mathrm{(E)}\ 16</math>
 
<math>\mathrm{(A)}\ 8\qquad \mathrm{(B)}\ 10\qquad \mathrm{(C)}\ 12\qquad \mathrm{(D)}\ 14\qquad \mathrm{(E)}\ 16</math>
  
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== Solution ==
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== Solution==
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The box P has dimensions <math>a</math>, <math>b</math>, and <math>c</math>. Therefore,
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Box P has dimensions <math>l</math>, <math>w</math>, and <math>h</math>.  
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<cmath>2ab+2ac+2bc=384</cmath>
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Surface area = <cmath>2lw+2lh+2wl=384</cmath>
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<cmath>4a+4b+4c=112 \Longrightarrow a + b + c = 28</cmath>
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Sum of all edges = <cmath>4l+4w+4h=112 \Longrightarrow l + w + h = 28</cmath>
  
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Now we make a formula for <math>r</math>. Since the [[diameter]] of the sphere is the space diagonal of the box,
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The diameter of the sphere is the space diagonal of the prism, which is <cmath>\sqrt{l^2 + w^2 +h^2}</cmath>
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<cmath>r=\frac{\sqrt{a^2+b^2+c^2}}{2}</cmath>
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<cmath>(l + w + h)^2 - (2lw + 2lh + 2wh) = l^2 + w^2 + h^2 = 784 - 384 = 400</cmath>
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<cmath>\sqrt{l^2 + w^2 +h^2} = 20 = diameter</cmath>
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We square <math>a+b+c</math>:
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<math>r=\frac{20}{2} = 10</math>$
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<cmath>(a+b+c)^2=a^2+b^2+c^2+2ab+2ac+2bc=a^2+b^2+c^2+384=784</cmath>
 
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We get that
 
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<cmath>r=\frac{\sqrt{a^2+b^2+c^2}}{2}=\boxed{\textbf{(B) }10}</cmath>
 
  
 
== See also ==
 
== See also ==

Revision as of 19:05, 19 June 2017

Problem

A rectangular box $P$ is inscribed in a sphere of radius $r$. The surface area of $P$ is 384, and the sum of the lengths of its 12 edges is 112. What is $r$?

$\mathrm{(A)}\ 8\qquad \mathrm{(B)}\ 10\qquad \mathrm{(C)}\ 12\qquad \mathrm{(D)}\ 14\qquad \mathrm{(E)}\ 16$

Solution

Box P has dimensions $l$, $w$, and $h$. Surface area = \[2lw+2lh+2wl=384\] Sum of all edges = \[4l+4w+4h=112 \Longrightarrow l + w + h = 28\]

The diameter of the sphere is the space diagonal of the prism, which is \[\sqrt{l^2 + w^2 +h^2}\] \[(l + w + h)^2 - (2lw + 2lh + 2wh) = l^2 + w^2 + h^2 = 784 - 384 = 400\] \[\sqrt{l^2 + w^2 +h^2} = 20 = diameter\] $r=\frac{20}{2} = 10$$

See also

2005 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 21 		Followed by
Problem 23
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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