Difference between revisions of "2023 AMC 12B Problems/Problem 13"
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Revision as of 19:43, 15 November 2023
- The following problem is from both the 2023 AMC 10B #17 and 2023 AMC 12B #13, so both problems redirect to this page.
Contents
[hide]Problem
A rectangular box P has distinct edge lengths , , and . The sum of the lengths of all edges of P is , 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 1 (algebraic manipulation)
We can create three equations using the given information. We also know that we want because that is the length that can be found from using the Pythagorean Theorem. We know that . . So . So our answer is .
Interestingly, we don't use the fact that the volume is . ~andliu766
~lprado ~Technodoggo ~minor edits and add-ons by lucaswujc
Solution 2 (factoring a polynomial)
We use the equations from Solution 1 and manipulate it a little: Notice how these are the equations for the vieta's formulas for a polynomial with roots of , , and . Let's create that polynomial. It would be . Multiplying each term by 4 to get rid of fractions, we get . Notice how the coefficients add up to . Whenever this happens, that means that is a factor and that 1 is a root. After using synthetic division to divide by , we get . Factoring that, you get . This means that this polynomial factors to and that the roots are , , and . Since we're looking for , this is equal to
~lprado
Solution 3 (find side lengths)
Let be the edge lengths.
Then, you can notice that these look like results of Vieta's formula: Finding when this will give us the edge lengths. We can use RRT to find one of the roots: One is , dividing gives . The other 2 roots are
Then, once we find the 3 edges being and , we can plug in to the distance formula to get .
-HIA2020
Solution 4 (Cheese Method)
Incorporating the solution above, the side lengths are larger than (a unit cube). The side length of the interior of a unit cube is , and we know that the side lengths are larger than , so that means the diagonal has to be larger than , and the only answer choice larger than
~kabbybear
See Also
2023 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 16 |
Followed by Problem 18 | |
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 10 Problems and Solutions |
2023 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 12 |
Followed by Problem 14 |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.