Difference between revisions of "2023 AMC 12B Problems/Problem 13"
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− | + | ==Problem== | |
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+ | A rectangular box P has distinct edge lengths <math>a</math>, <math>b</math>, and <math>c</math>. The sum of the lengths of all <math>12</math> edges of P is <math>13</math>, the areas of all 6 faces of P is <math>\frac{11}{2}</math>, and the volume of P is <math>\frac{1}{2}</math>. What is the length of the longets interior diagonal connecting two vertices of P? | ||
+ | |||
+ | ==Solution 1 (algebraic manipulation)== | ||
+ | We can create three equationss using the given information. | ||
+ | <cmath>4a+4b+4c = 13</cmath> | ||
+ | <cmath>2ab+2ac+2bc=\frac{11}{2}</cmath> | ||
+ | <cmath>abc=\frac{1}{2}</cmath> | ||
+ | We also know that we want <math>\sqrt{a^2 + b^2 + c^2}</math>. We know that <math>a^2 + b^2 + c^2 = (a+b+c)^2 - 2ab - 2ac - 2bc</math>. <math>a+b+c = \frac{13}{4}</math>. So <math>a^2 + b^2 + c^2 = (\frac{13}{4})^2 - \frac{11}{2} = \frac{169-88}{16} = \frac{81}{16}</math>. So our answer is <math>\sqrt{\frac{81}{16}} = \boxed{\frac{9}{4}}</math>. |
Revision as of 15:21, 15 November 2023
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 longets interior diagonal connecting two vertices of P?
Solution 1 (algebraic manipulation)
We can create three equationss using the given information.
We also know that we want
. We know that
.
. So
. So our answer is
.