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

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== Problem ==
 
== Problem ==
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A rectangular box <math> P </math> is [[inscribe]]d in a [[sphere]] of [[radius]] <math>r</math>. The [[surface area]] of <math>P</math> is 384, and the sum of the lengths of it's 12 edges is 112. What is <math>r</math>?
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A rectangular box <math> P </math> is inscribed in a sphere of radius <math>r</math>. The surface area of <math>P</math> is 384, and the sum of the lengths of its 12 edges is 112. What is <math>r</math>?
  
 
<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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<cmath>2ab+2ac+2bc=384</cmath>
 
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<cmath>4a+4b+4c=112 \Longrightarrow a + b + c = 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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Box P has dimensions <math>l</math>, <math>w</math>, and <math>h</math>.  
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<cmath>r=\frac{\sqrt{a^2+b^2+c^2}}{2}</cmath>
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Its surface area is <cmath>2lw+2lh+2wh=384,</cmath>
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and the sum of all its edges is <cmath>l + w + h = \dfrac{4l+4w+4h}{4} = \dfrac{112}{4} = 28.</cmath>
  
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We square <math>a+b+c</math>:
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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>(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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Notice that <cmath>(l + w + h)^2 - (2lw + 2lh + 2wh) = l^2 + w^2 + h^2 = 784 - 384 = 400,</cmath> so the diameter is
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<cmath>\sqrt{l^2 + w^2 +h^2} = \sqrt{400} = 20.</cmath> The radius is half of the diameter, so
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<cmath>r=\frac{20}{2} = \boxed{\textbf{(B)} 10}.</cmath>
  
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We get that
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== Solution 2 ==
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<cmath>r=\frac{\sqrt{a^2+b^2+c^2}}{2}=B)10</cmath>
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As in the previous solution, we have that <math>2lw+2lh+2wh=384</math> and <math>l + w + h = \dfrac{4l+4w+4h}{4} = \dfrac{112}{4} = 28</math>, and the diameter of the sphere is the space diagonal of the prism, <math>\sqrt{l^2 + w^2 + h^2}</math>.
 +
 
 +
 
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Now, since this is competition math, we only need to find the space diagonal of any one box that fits the requirements, so assume that <math>h=0</math>. (This essentially means that we have an infinitesimally thin box.) We now have that <math>2lw = 384</math> and <math>l + w = 28</math>, and we are solving for <math>\sqrt{l^2 + w^2}</math>. Because <cmath>(l + w)^2 - 2lw = l^2 + 2lw + w^2 - 2lw = l^2 + w^2,</cmath> this means that <cmath>l^2 + w^2 = 28^2 - 384 = 400,</cmath> so the space diagonal is <math>\sqrt{400} = 20</math>. Since the diameter of the sphere is <math>20</math>, the radius is <math>\boxed{\textbf{(B) } 10}</math>. ~[[User:emerald_block|emerald_block]]
  
 
== See also ==
 
== See also ==

Latest revision as of 09:20, 16 September 2022

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 $h$. Its surface area is \[2lw+2lh+2wh=384,\] and the sum of all its edges is \[l + w + h = \dfrac{4l+4w+4h}{4} = \dfrac{112}{4} = 28.\]

The diameter of the sphere is the space diagonal of the prism, which is \[\sqrt{l^2 + w^2 +h^2}.\] Notice that \[(l + w + h)^2 - (2lw + 2lh + 2wh) = l^2 + w^2 + h^2 = 784 - 384 = 400,\] so the diameter is \[\sqrt{l^2 + w^2 +h^2} = \sqrt{400} = 20.\] The radius is half of the diameter, so \[r=\frac{20}{2} = \boxed{\textbf{(B)} 10}.\]

Solution 2

As in the previous solution, we have that $2lw+2lh+2wh=384$ and $l + w + h = \dfrac{4l+4w+4h}{4} = \dfrac{112}{4} = 28$, and the diameter of the sphere is the space diagonal of the prism, $\sqrt{l^2 + w^2 + h^2}$.


Now, since this is competition math, we only need to find the space diagonal of any one box that fits the requirements, so assume that $h=0$. (This essentially means that we have an infinitesimally thin box.) We now have that $2lw = 384$ and $l + w = 28$, and we are solving for $\sqrt{l^2 + w^2}$. Because \[(l + w)^2 - 2lw = l^2 + 2lw + w^2 - 2lw = l^2 + w^2,\] this means that \[l^2 + w^2 = 28^2 - 384 = 400,\] so the space diagonal is $\sqrt{400} = 20$. Since the diameter of the sphere is $20$, the radius is $\boxed{\textbf{(B) } 10}$. ~emerald_block

See also

2005 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 21 		Followed by
Problem 23
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All AMC 12 Problems and Solutions

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