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Difference between revisions of "2023 AMC 12B Problems/Problem 13"

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m (→‎Problem: use $\mathcal{P}$ instead of $P$, as in the original problem's wording)
 
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==Problem==
 
==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 longest interior diagonal connecting two vertices of P?
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A rectangular box <math>\mathcal{P}</math> 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 <math>\mathcal{P}</math> is <math>13</math>, the areas of all <math>6</math> faces of <math>\mathcal{P}</math> is <math>\frac{11}{2}</math>, and the volume of <math>\mathcal{P}</math> is <math>\frac{1}{2}</math>. What is the length of the longest interior diagonal connecting two vertices of <math>\mathcal{P}</math>?
 +
 
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<math>\textbf{(A)}~2\qquad\textbf{(B)}~\frac{3}{8}\qquad\textbf{(C)}~\frac{9}{8}\qquad\textbf{(D)}~\frac{9}{4}\qquad\textbf{(E)}~\frac{3}{2}</math>
 +
 
 +
==Video Solution by MegaMath==
 +
 
 +
https://www.youtube.com/watch?v=le0KSx3Cy-g&t=28s
  
 
==Solution 1 (algebraic manipulation)==
 
==Solution 1 (algebraic manipulation)==
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<cmath>2ab+2ac+2bc=\frac{11}{2}</cmath>
 
<cmath>2ab+2ac+2bc=\frac{11}{2}</cmath>
 
<cmath>abc=\frac{1}{2}</cmath>
 
<cmath>abc=\frac{1}{2}</cmath>
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We also know that we want <math>\sqrt{a^2 + b^2 + c^2}</math> because that is the length that can be found from using the Pythagorean Theorem. 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 = \left(\frac{13}{4}\right)^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>.
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We also know that we want <math>\sqrt{a^2 + b^2 + c^2}</math> because that is the length that can be found from using the Pythagorean Theorem. We cleverly notice that <math>a^2 + b^2 + c^2 = (a+b+c)^2 - 2(ab+ac+bc)</math>. We know that <math>a+b+c = \frac{13}{4}</math> and <math>2(ab+ac+bc)=\dfrac{11}2</math>, so <math>a^2 + b^2 + c^2 = \left(\frac{13}{4}\right)^2 - \frac{11}{2} = \frac{169-88}{16} = \frac{81}{16}</math>. So our answer is <math>\sqrt{\frac{81}{16}} = \boxed{\textbf{(D)}~\tfrac94}</math>.
  
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Interestingly, we don't use the fact that the volume is <math>\frac{1}{2}</math>. ~andliu766
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Interestingly, we don't use the fact that the volume is <math>\frac{1}{2}</math>.
  
 
~lprado
 
~lprado
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~Technodoggo
 
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~minor edits and add-ons by lucaswujc
 
  
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==Solution 2 (factoring a polynomial)==
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~minor edits and add-ons by Technodoggo, lucaswujc, andliu766, and BcMath
 +
 
 +
==Solution 2 (Vieta's)==
 
We use the equations from Solution 1 and manipulate it a little:
 
We use the equations from Solution 1 and manipulate it a little:
 
<cmath>a+b+c = \frac{13}{4}</cmath>
 
<cmath>a+b+c = \frac{13}{4}</cmath>
 
<cmath>ab+ac+bc=\frac{11}{4}</cmath>
 
<cmath>ab+ac+bc=\frac{11}{4}</cmath>
 
<cmath>abc=\frac{1}{2}</cmath>
 
<cmath>abc=\frac{1}{2}</cmath>
−
Notice how these are the equations for the vieta's formulas for a polynomial with roots of <math>a</math>, <math>b</math>, and <math>c</math>. Let's create that polynomial. It would be <math>x^3 - \frac{13}{4}x^2 + \frac{11}{4}x - \frac{1}{2}</math>. Multiplying each term by 4 to get rid of fractions, we get <math>4x^3 - 13x^2 + 11x - 2</math>. Notice how the coefficients add up to <math>0</math>. Whenever this happens, that means that <math>(x-1)</math> is a factor and that 1 is a root. After using synthetic division to divide <math>4x^3 - 13x^2 + 11x - 2</math> by <math>x-1</math>, we get <math>4x^2 - 9x + 2</math>. Factoring that, you get <math>(x-2)(4x-1)</math>. This means that this polynomial factors to <math>(x-1)(x-2)(4x-1)</math> and that the roots are <math>1</math>, <math>2</math>, and <math>1/4</math>. Since we're looking for <math>\sqrt{a^2 + b^2 + c^2}</math>, this is equal to <math>\sqrt{1^2 + 2^2 + \frac{1}{4}^2} = \sqrt{\frac{81}{16}} = \boxed{\frac{9}{4}}</math>
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Notice how these are the equations for the vieta's formulas for a polynomial with roots of <math>a</math>, <math>b</math>, and <math>c</math>. Let's create that polynomial. It would be <math>x^3 - \frac{13}{4}x^2 + \frac{11}{4}x - \frac{1}{2}</math>. Multiplying each term by 4 to get rid of fractions, we get <math>4x^3 - 13x^2 + 11x - 2</math>. Notice how the coefficients add up to <math>0</math>. Whenever this happens, that means that <math>(x-1)</math> is a factor and that 1 is a root. After using synthetic division to divide <math>4x^3 - 13x^2 + 11x - 2</math> by <math>x-1</math>, we get <math>4x^2 - 9x + 2</math>. Factoring that, you get <math>(x-2)(4x-1)</math>. This means that this polynomial factors to <math>(x-1)(x-2)(4x-1)</math> and that the roots are <math>1</math>, <math>2</math>, and <math>1/4</math>. Since we're looking for <math>\sqrt{a^2 + b^2 + c^2}</math>, this is equal to <math>\sqrt{1^2 + 2^2 + \frac{1}{4}^2} = \sqrt{\frac{81}{16}} = \boxed{\textbf{(D)}~\tfrac94}</math>
  
 
~lprado
 
~lprado
  
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==Solution 3 (find side lengths)==
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==Solution 3 (Cheese Method)==
  
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Let <math>a,b,c</math> be the edge lengths.
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Incorporating the solution above, we know <math>a+b+c</math> = <math>13/4</math> <math>\Rightarrow</math> <math>a+b+c > 3</math>. The side lengths are larger than <math>1</math> <math>\cdot</math> <math>1</math> <math>\cdot</math> <math>1</math> (a unit cube). The side length of the interior of a unit cube is <math>\sqrt{3}</math>, and we know that the side lengths are larger than <math>1</math> <math>\cdot</math> <math>1</math> <math>\cdot</math> <math>1</math>, so that means the diagonal has to be larger than <math>\sqrt{3}</math>, and the only answer choice larger than <math>\sqrt{3}</math> <math>\Rightarrow</math> <math>\boxed{\textbf{(D)}~\tfrac94}</math>
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<math>4(a+b+c)=13, a+b+c=13/4</math>
 
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<math>2(ab+bc+ac)=11/2, ab+bc+ac=11/4</math>
 
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<math>abc=1/2</math>
 
  
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Then, you can notice that these look like results of Vieta's formula:
 
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<math>(x-a)(x-b)(x-c) = x^3-(a+b+c)x^2+(ab+bc+ac)x-abc = x^3-13/4x^2+11/4x-1/2</math>
 
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Finding when this <math>= 0</math> will give us the edge lengths.
 
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We can use RRT to find one of the roots:
 
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One is <math>x=1</math>, dividing gives <math>x^2-9/4x+1/2</math>.
 
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The other 2 roots are <math>2,1/4</math>
 
  
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Then, once we find the 3 edges being <math>a=1,b=2,</math> and <math>c=1/4</math>, we can plug in to the distance formula to get <math>9/4</math>.
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~kabbybear
  
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Note that the real number <math>\sqrt{3}</math> is around <math>1.73</math>. Option <math>A</math> is also greater than <math>\sqrt{3}</math> meaning there are two options greater than <math>\sqrt{3}</math>. Option <math>A</math> is an integer so educationally guessing we arrive at answer <math>D</math> <math>\Rightarrow</math> <math>\boxed{\textbf{(D)}~\tfrac94}</math>
  
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-HIA2020
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~atictacksh
  
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==Solution 4 (Cheese Method)==
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==Video Solution 1 by OmegaLearn==
 +
https://youtu.be/bXbOPnIAKPo
  
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Incorporating the solution above, the side lengths are larger than <math>1</math> <math>\cdot</math> <math>1</math> <math>\cdot</math> <math>1</math> (a unit cube). The side length of the interior of a unit cube is <math>\sqrt{3}</math>, and we know that the side lengths are larger than <math>1</math> <math>\cdot</math> <math>1</math> <math>\cdot</math> <math>1</math>, so that means the diagonal has to be larger than <math>\sqrt{3}</math>, and the only answer choice larger than <math>\sqrt{3}</math> <math>\Rightarrow</math> <math>\boxed {\textbf{(D) 9/4}}</math>
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==Video Solution 2 by SpreadTheMathLove==
 +
https://www.youtube.com/watch?v=lVkvcCmY9uM
  
 +
==Video Solution==
  
−
~kabbybear
+
https://youtu.be/4jjWyikA7mg
  
 +
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
  
 
==See Also==
 
==See Also==

Latest revision as of 17:02, 11 September 2024

The following problem is from both the 2023 AMC 10B #17 and 2023 AMC 12B #13, so both problems redirect to this page.

Problem

A rectangular box $\mathcal{P}$ has distinct edge lengths $a$, $b$, and $c$. The sum of the lengths of all $12$ edges of $\mathcal{P}$ is $13$, the areas of all $6$ faces of $\mathcal{P}$ is $\frac{11}{2}$, and the volume of $\mathcal{P}$ is $\frac{1}{2}$. What is the length of the longest interior diagonal connecting two vertices of $\mathcal{P}$?

$\textbf{(A)}~2\qquad\textbf{(B)}~\frac{3}{8}\qquad\textbf{(C)}~\frac{9}{8}\qquad\textbf{(D)}~\frac{9}{4}\qquad\textbf{(E)}~\frac{3}{2}$

Video Solution by MegaMath

https://www.youtube.com/watch?v=le0KSx3Cy-g&t=28s

Solution 1 (algebraic manipulation)

[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]

We can create three equations using the given information. \[4a+4b+4c = 13\] \[2ab+2ac+2bc=\frac{11}{2}\] \[abc=\frac{1}{2}\] We also know that we want $\sqrt{a^2 + b^2 + c^2}$ because that is the length that can be found from using the Pythagorean Theorem. We cleverly notice that $a^2 + b^2 + c^2 = (a+b+c)^2 - 2(ab+ac+bc)$. We know that $a+b+c = \frac{13}{4}$ and $2(ab+ac+bc)=\dfrac{11}2$, so $a^2 + b^2 + c^2 = \left(\frac{13}{4}\right)^2 - \frac{11}{2} = \frac{169-88}{16} = \frac{81}{16}$. So our answer is $\sqrt{\frac{81}{16}} = \boxed{\textbf{(D)}~\tfrac94}$.

Interestingly, we don't use the fact that the volume is $\frac{1}{2}$.

~lprado

~minor edits and add-ons by Technodoggo, lucaswujc, andliu766, and BcMath

Solution 2 (Vieta's)

We use the equations from Solution 1 and manipulate it a little: \[a+b+c = \frac{13}{4}\] \[ab+ac+bc=\frac{11}{4}\] \[abc=\frac{1}{2}\] Notice how these are the equations for the vieta's formulas for a polynomial with roots of $a$, $b$, and $c$. Let's create that polynomial. It would be $x^3 - \frac{13}{4}x^2 + \frac{11}{4}x - \frac{1}{2}$. Multiplying each term by 4 to get rid of fractions, we get $4x^3 - 13x^2 + 11x - 2$. Notice how the coefficients add up to $0$. Whenever this happens, that means that $(x-1)$ is a factor and that 1 is a root. After using synthetic division to divide $4x^3 - 13x^2 + 11x - 2$ by $x-1$, we get $4x^2 - 9x + 2$. Factoring that, you get $(x-2)(4x-1)$. This means that this polynomial factors to $(x-1)(x-2)(4x-1)$ and that the roots are $1$, $2$, and $1/4$. Since we're looking for $\sqrt{a^2 + b^2 + c^2}$, this is equal to $\sqrt{1^2 + 2^2 + \frac{1}{4}^2} = \sqrt{\frac{81}{16}} = \boxed{\textbf{(D)}~\tfrac94}$

~lprado

Solution 3 (Cheese Method)

Incorporating the solution above, we know $a+b+c$ = $13/4$ $\Rightarrow$ $a+b+c > 3$. The side lengths are larger than $1$ $\cdot$ $1$ $\cdot$ $1$ (a unit cube). The side length of the interior of a unit cube is $\sqrt{3}$, and we know that the side lengths are larger than $1$ $\cdot$ $1$ $\cdot$ $1$, so that means the diagonal has to be larger than $\sqrt{3}$, and the only answer choice larger than $\sqrt{3}$ $\Rightarrow$ $\boxed{\textbf{(D)}~\tfrac94}$


~kabbybear

Note that the real number $\sqrt{3}$ is around $1.73$. Option $A$ is also greater than $\sqrt{3}$ meaning there are two options greater than $\sqrt{3}$. Option $A$ is an integer so educationally guessing we arrive at answer $D$ $\Rightarrow$ $\boxed{\textbf{(D)}~\tfrac94}$

~atictacksh

Video Solution 1 by OmegaLearn

https://youtu.be/bXbOPnIAKPo

Video Solution 2 by SpreadTheMathLove

https://www.youtube.com/watch?v=lVkvcCmY9uM

Video Solution

https://youtu.be/4jjWyikA7mg

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

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

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