Difference between revisions of "1996 AHSME Problems/Problem 23"

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==Problem==
 
==Problem==
 
 
The sum of the lengths of the twelve edges of a rectangular box is <math>140</math>, and  
 
The sum of the lengths of the twelve edges of a rectangular box is <math>140</math>, and  
 
the distance from one corner of the box to the farthest corner is <math>21</math>. The total  
 
the distance from one corner of the box to the farthest corner is <math>21</math>. The total  
surface area of the box is  
+
[[surface area]] of the box is  
  
 
<math> \text{(A)}\ 776\qquad\text{(B)}\ 784\qquad\text{(C)}\ 798\qquad\text{(D)}\ 800\qquad\text{(E)}\ 812 </math>
 
<math> \text{(A)}\ 776\qquad\text{(B)}\ 784\qquad\text{(C)}\ 798\qquad\text{(D)}\ 800\qquad\text{(E)}\ 812 </math>
  
 
==Solution==
 
==Solution==
 
+
Let <math>x, y</math>, and <math>z</math> be the unique lengths of the edges of the box.  Each box has <math>4</math> edges of each length, so:
Let <math>x</math>, <math>y</math>, and <math>z</math> be the unique lengths of the edges of the box.  Each box has <math>4</math> edges of each length, so:
+
<cmath>4x + 4y + 4z = 140 \ \Longrightarrow \ x + y + z = 35.</cmath>
 
+
The spacial diagonal (longest distance) is given by <math>\sqrt{x^2 + y^2 + z^2}</math>.  Thus, we have <math>\sqrt{x^2 + y^2 + z^2} = 21</math>, so <math>x^2 + y^2 + z^2 = 21^2</math>.
<math>4x + 4y + 4z = 140</math>
 
 
 
<math>x + y + z = 35</math>
 
 
 
The spacial diagonal (longest distance) is given by <math>\sqrt{x^2 + y^2 + z^2}</math>.  Thus, we have:
 
 
 
<math>\sqrt{x^2 + y^2 + z^2} = 21</math>
 
 
 
<math>x^2 + y^2 + z^2 = 21^2</math>
 
  
 
Our target expression is the surface area of the box:
 
Our target expression is the surface area of the box:
  
<math>S = 2xy + 2xz + 2yz</math>
+
<cmath>S = 2xy + 2xz + 2yz.</cmath>
  
Since <math>S</math> is a symmetric polynomial of degree <math>2</math>, we try squaring the first equation to get:
+
Since <math>S</math> is a [[Elementary symmetric sum|symmetric polynomial]] of degree <math>2</math>, we try squaring the first equation to get:
  
<math>(x + y + z)^2 = 35^2</math>
+
<cmath>35^2 = (x + y + z)^2 = x^2 + y^2 + z^2 + 2xy +2yz + 2xz = 35^2.</cmath>
  
<math>x^2 + y^2 + z^2 + 2xy +2yz + 2xz = 35^2</math>
+
Substituting in our long diagonal and surface area expressions, we get: <math>21^2 + S = 35^2</math>, so <math>S = (35 + 21)(35 - 21) = 56\cdot 14 = 784</math>, which is option <math>\boxed{(\text{B})}</math>.
 
 
Substituting in our long diagonal and surface area expressions, we get:
 
 
 
<math>21^2 + S = 35^2</math>
 
 
 
<math>S = (35 + 21)(35 - 21)</math>
 
 
 
<math>S = 56\cdot 14</math>
 
 
 
<math>S = 784</math>, which is option <math>\boxed{B}</math>
 
  
 
==See also==
 
==See also==
 
{{AHSME box|year=1996|num-b=22|num-a=24}}
 
{{AHSME box|year=1996|num-b=22|num-a=24}}
 +
 +
[[Category:Introductory Algebra Problems]]
 +
[[Category:Algebraic Manipulations Problems]]

Revision as of 16:00, 20 August 2011

Problem

The sum of the lengths of the twelve edges of a rectangular box is $140$, and the distance from one corner of the box to the farthest corner is $21$. The total surface area of the box is

$\text{(A)}\ 776\qquad\text{(B)}\ 784\qquad\text{(C)}\ 798\qquad\text{(D)}\ 800\qquad\text{(E)}\ 812$

Solution

Let $x, y$, and $z$ be the unique lengths of the edges of the box. Each box has $4$ edges of each length, so: \[4x + 4y + 4z = 140 \ \Longrightarrow \ x + y + z = 35.\] The spacial diagonal (longest distance) is given by $\sqrt{x^2 + y^2 + z^2}$. Thus, we have $\sqrt{x^2 + y^2 + z^2} = 21$, so $x^2 + y^2 + z^2 = 21^2$.

Our target expression is the surface area of the box:

\[S = 2xy + 2xz + 2yz.\]

Since $S$ is a symmetric polynomial of degree $2$, we try squaring the first equation to get:

\[35^2 = (x + y + z)^2 = x^2 + y^2 + z^2 + 2xy +2yz + 2xz = 35^2.\]

Substituting in our long diagonal and surface area expressions, we get: $21^2 + S = 35^2$, so $S = (35 + 21)(35 - 21) = 56\cdot 14 = 784$, which is option $\boxed{(\text{B})}$.

See also

1996 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 22
Followed by
Problem 24
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