2023 AMC 10B Problems/Problem 17
Contents
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 1
Let and be the sides of the box, we get
The diagonal of the box is
~Technodoggo
Note
Interestingly, we don't use the fact that the volume is ~andliu766
Solution 2 (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 3 (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