Difference between revisions of "2023 AMC 12B Problems/Problem 13"
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Revision as of 20:29, 15 November 2023
Contents
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
.
~lprado
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
See Also
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.