Difference between revisions of "1996 AHSME Problems/Problem 23"
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==Problem== | ==Problem== | ||
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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 | + | <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>. | |
| − | < | ||
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| − | |||
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| − | 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: | ||
| − | < | + | <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: |
| − | < | + | <cmath>35^2 = (x + y + z)^2 = x^2 + y^2 + z^2 + 2xy +2yz + 2xz = 35^2.</cmath> |
| − | + | 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) | ||
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==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 15:00, 20 August 2011
Problem
The sum of the lengths of the twelve edges of a rectangular box is 
, and
the distance from one corner of the box to the farthest corner is 
. The total
surface area of the box is
Solution
Let 
, and 
be the unique lengths of the edges of the box. Each box has 
edges of each length, so:
![\[4x + 4y + 4z = 140 \ \Longrightarrow \ x + y + z = 35.\]](http://latex.artofproblemsolving.com/b/e/2/be25f71f94650c57a8ba91c53686a16502816e25.png)
The spacial diagonal (longest distance) is given by 
. Thus, we have 
, so 
.
Our target expression is the surface area of the box:
![\[S = 2xy + 2xz + 2yz.\]](http://latex.artofproblemsolving.com/f/d/6/fd64bf92913770b8d02c494364ff550659183689.png)
Since 
is a symmetric polynomial of degree 
, we try squaring the first equation to get:
![\[35^2 = (x + y + z)^2 = x^2 + y^2 + z^2 + 2xy +2yz + 2xz = 35^2.\]](http://latex.artofproblemsolving.com/5/f/9/5f9a31cb8b0f4c0deb30ba3b528cab5084e1f448.png)
Substituting in our long diagonal and surface area expressions, we get: 
, so 
, which is option 
.
See also
| 1996 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 22 |
Followed by Problem 24 | |
| 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 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
