Difference between revisions of "2023 AMC 10B Problems/Problem 17"
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| + | == Problem == | ||
| + | A rectangular box 𝒫 has distinct edge lengths 𝑎, 𝑏, and 𝑐. The sum of the lengths of | ||
| + | all 12 edges of 𝒫 is 13, the sum of the areas of all 6 faces of 𝒫 is <math>\dfrac{11}{2}</math>, and the volume of 𝒫 is <math>\dfrac{1}{2}</math>. What is the length of the longest interior diagonal connecting two vertices of 𝒫 ? | ||
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== Solution == | == Solution == | ||
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<asy> | <asy> | ||
Revision as of 15:37, 15 November 2023
Problem
A rectangular box 𝒫 has distinct edge lengths 𝑎, 𝑏, and 𝑐. The sum of the lengths of
all 12 edges of 𝒫 is 13, the sum of the areas of all 6 faces of 𝒫 is 
, and the volume of 𝒫 is 
. What is the length of the longest interior diagonal connecting two vertices of 𝒫 ?
Solution
![[asy] import geometry; pair A = (-3, 4); pair B = (-3, 5); pair C = (-1, 4); pair D = (-1, 5); pair AA = (0, 0); pair BB = (0, 1); pair CC = (2, 0); pair DD = (2, 1); draw(D--AA,dashed); draw(A--B); draw(A--C); draw(B--D); draw(C--D); draw(A--AA); draw(B--BB); draw(C--CC); draw(D--DD); // Dotted vertices dot(A); dot(B); dot(C); dot(D); dot(AA); dot(BB); dot(CC); dot(DD); draw(AA--BB); draw(AA--CC); draw(BB--DD); draw(CC--DD); label("a",midpoint(D--DD),E); label("b",midpoint(CC--DD),E); label("c",midpoint(AA--CC),S); [/asy]](http://latex.artofproblemsolving.com/4/e/4/4e4fcd74b454deedf3f32aa14c6c0e53b07d3002.png)
Let 
and 
be the sides of the box, we get
The diagonal of the box is
~Technodoggo
